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·SOLID GEOMETRY

·A SERIES OF MATHEMATICAL TEXTS
EDITED BY
EARLE RAYMOND HEDRICK
THE CALCULUS
By ELLERY WILLIAMS Davis and WilLIAM CHARLES
BRENKE.
PLANE AND SOLIT) ANALYTIC GEOMETRY
By ALEXANDER ZIWET and Louis ALLEN HOPKINS.
PLANE ANI) SPHERICAL TRIGONOMETRY WITH
COMPLETE TABLES
By ALFRED MONROE KENYON and LOUIS INGOLD.
PLANE AND SPHERICAL TRIGONOMETRY WITH
BRIEF TABLES
By ALFRED MONROE KENYON and Louis INGOLD.
THE MACMILLAN TABLES
Prepared under the direction of EARLE RAYMOND HEDRICK.
PLANE GEOMETRY
By WALTER Burton FORD and CHARLES AMMERMAN.
PLANE AND SOLID GEOMETRY
By WALTER BURTON Ford and CHARLES AMMERMAN.
SOLID GEOMETRY
By Walter BURTON Ford and CHARLES AMMERMAN.

·SOLID GEOMETRY
BY
WALTER BURTON FORD
JUNIOR PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF MICHIGAN
AND
CHARLES AMMERMAN
THE WILLIAM MCKINLEY HIGH SCHOOL, ST. LOUIS
EDITED BY
EARLE RAYMOND HEDRICK
*
New York
THE MACMILLAN COMPANY
1920
All rights reserved

·COPYRIGAT, 1913,
BY THE MACMILLAN COMPANY.
Set up and electrotyped. Published November, 1913.
Norwood Press
J. 8. Cushing Co. -- Berwick & Smith Co.
Norwood, Mass., U.S.A.

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·vi
PREFACE
6
The great excellence of the figures, particularly the very
unusual and effective phantom' halftone engravings, deserves
mention. These figures should go far toward relieving the
unreality which often attaches to the constructions of Solid
Geometry in the minds of students.
W. B. FORD,
CHAS. AMMERMAN,
E. R. TIEDRICK, EDITOR.

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·CONTENTS PAGT8 215-237 Chapter VI. Lines and Planes in Space. Part I. General Principles Part II. Perpendiculars and Parallels Part III. Dihedral Angles Part IV. Polyhedral Angles Miscellaneous Exercises on Chapter VI 215-218 219-227 228-231 232-234 235-237 Chapter VII. Polyhedrons. Cylinders. Cones Part I. Prisms. Parallelepipeds Part II. Pyramids Part III. Cylinders and Cones Part IV. General Theorems on Polyhedrons. Similarity. Regular Solids. Volumes Miscellaneous Exercises on Chapter VII 238-283 238-252 253-262 263-273 274-281 282-283 . Chapter VIII. The Sphere Part I. General Properties Part II. Spherical Angles. Triangles. Polygons Part III. Areas and Volumes Miscellaneous Exercises on Chapter VIII 284-321 284-293 296-308 309-317 317-321 . Tables Table I. Quantities Determined by a Given Angle. Table II. Powers and Roots Table III. Important Numbers i-xxvii ii-vii viii-xxvi xxvii Syllabus of Plane Geometry Introduction. Chapter I. Rectilinear Figures Chapter II. The Circle viii xxix-xlvi xxix xxxii xxxvii ·CONTENTS IX xxxix . Chapter III. Proportion. Similarity Chapter IV. Areas of Polygons. Pythagorean Theorem Chapter V. Regular Polygons and Circles Appendix : Maxima and Minima xliii xliv xly Index xlvii-xlix The complete text for the theorems listed in the Syllabus of Theorems of Plane Geometry in this book is published separately under the title PLANE GEOMETRY The contents of that book and of this one are published together under the title PLANE AND SOLID GEOMETRY *

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·SOLID GEOMETRY CHAPTER VI LINES AND PLANES IN SPACE PART I. GENERAL PRINCIPLES 239. Definitions. Solid Geometry, or Geometry of Three Dimensions, treats of figures whose parts are not confined to a plane. A plane is a surface such that if any two points in it are taken, the straight line passing through them lies wholly in the surface. M A B N FIG. 103 Thus, in Fig. 163, if A and B are two points of a plane MN, the entire straight line AB lies in the plane MN. Any point 0 on AB lies in MN. A plane is said to be determined by certain points and lines if that plane and no other plane contains those points and lines, 240. Corollary 1. It is evident from the definition of a plane that if a line has two of its points in a plane, it lies wholly in that plane. 215 ·216 LINES AND PLANES IN SPACE [VI, S 241 241. Assumptions, or Postulates. 1. A plane is unlimited in extent. 2. Through any straight line an unlimited number of planes may be passed. See Fig. 164. 3. If a plane is revolved about any straight line in it as an axis, it may be inaile to pass through FIG. 164 any point in space. 4. One and only one plane can be made to pass through three points not in the same straight line. C P •A B* FIG. 165 (a) FIG. 165 (6) Fig. 165 (a) represents a plane PQ througlı three points A, B, C. Fig. 165 (b) represents a plane piece of glass resting on the points of three tacks. 5. Two planes cannot intersect each other in only a single point. 242. Corollary 1. A plane is letermined by two intersecting lines. M N A B FIG. 166 (a) FIG. 166 (6) [HINT. Consider the point where the lines intersect, and two other points, one on each line; then apply 4, § 241.]

·VI, S 244) GENERAL PRINCIPLES 217 A line and a point without the line de- 243. Corollary 2. termine a plane. [Hint. Use 4, § 241.] M N FIG. 167 (1) FIG. 167 (6) 244. Corollary 3. Two parallel lincs determine a plane. M N B F16. 168 (a) Fig. 168 (6) [IIInt. By the lefinition of parallel lines ($ 18), two such lines must lie in a plane. Show that this is the only one.] EXERCISES 1. How many planes pass through a given straight line in space? How many pass through two given points ? 2. In a carpenter's plane the knife-edge lies along a straight line. As soon as any rough surface has been sufficiently planed off, the whole length of the knife-edge keeps on the surface as the plane is moved along. Connect this fact with $ 240. 3. Why are cameras, surveyors' transits, etc., mounted on three legs instead of four ? 4. Prove that a straight line can intersect a plane in but one point unless it lies wholly in the plane. See $ 240.

·218 LINES AND PLANES IN SPACE (VI, S. 245 The intersection of two planes is (1 straight 245. Theorem I. line. Given the two intersecting planes MN and RS. To prove that their intersection is a straight line. Proof. Let A and B be any two points common to both planes. 5, $ 2:11 FIG. 109 Draw the straight line AB. Then every point in AB lies in MN and also in RS. § 240 Therefore, AB is common to the two planes. Moreover, no point not on AB can be common to both planes, for the two planes would then coincide. 4, § 241 Therefore, the intersection of the planes MN and RS is a straight line. EXERCISES 1. What is the locus of all points common to two intersecting planes ? 2. If a sheet of paper is folded, why is the crease straight? ! 3. In how many points (in general) will three planes intersect? What can be said of the intersection of four or more planes in space ? 4. Can two pencils he held in such a position that a plane cannot be passed through them? State the general fact about a plane that is illustrated by your answer. 5. Can a plane be passed (in general) through four or more given points in space ? Can a plane be passed (in general) through three lines all of which pass through a common point in space? 6. Can there be two straight lines that are not parallel and that do not meet? Find a pair of such lines in Fig. 169.

·VI, $ 248] PERPENDICULARS AND PARALLELS 219 PART II. PERPENDICULARS AND PARALLELS 246. Line Perpendicular to a Plane. The point where a line intersects a plane is called the foot of the line on that plane. A straight line is perpendicular to a plane when it is perpendicular to every straight line in the plane drawn through its Р I D B A N M E G Fig. 170 (c) Fig. 170 (0) foot. The plane is then also said to be perpendicular to the line. Thus, in Fig. 170 («), if PQ is perpendicular to the plane MN, it is then perpendicular to all the lines Q4, QB, QC, etc.; and PQ is called the distance from P to MN; see Ex. 1 below. 247. Parallel Planes and Lines. A straight line is parallel to a plane if they never meet, however far produced. Two planes are parallel if they never meet, however far produced. It is to be remembered ($ 48) that two lines are parallel only when they lie in the supie plane and do not meet. 248. Corollary 1. A plane that contains one of two parallel lines is parallel to the other line. EXERCISES 1. Show, by $ 77, that the perpendicular from a point P to a plane MN (Fig. 170 a) is shorter than any other line that can be drawn from P to MN. 2. Show, by $ 71, that if two oblique lines from a point P. to a plane MN cut off equal distances from the foot of the perpendicular from P to MN, they are equal. See Ex. 1, p. 63.

·220 LINES AND PLANES IN SPACE (VI, $ 249 249. Theorem II. If a line is perpendicular to each of two lines at their point of intersection, it is perpendicular to their plane. Given FB perpendicular at B to each of two straight lines AB and BC of the plane MN. To prove FB perpendicular to the plane MN. Proof. Draw AC, and through B draw any line, as BH, meeting AC at H. FIG. 171 Prolong FB to E so that BE=FB. Join F and E to A, II, and C. Then AB and BC are perpendicular bisectors of FE, Const. whence FA= AE, FC=CE. Therefore Δ AFC ΝΔ ΛΕC, whence LIAI ZILE. Why? Also AHAF A HAE, Why? and HF IIE. Why? Hence HBI FE or FB. Why? But HB was any line in MN drawn through B. Therefore FBI MN. $ 246 $ 100 250. Corollary 1. At a point in a plane only one perpendicular line can be erected. AIC [Hint. Suppose a second perpendicular line BC could be erected (Fig. 172). Pass a plane through AB and E N BC. This plane will intersect MN in a M B straight line, as DE. Then AB and BC are both perpendicular to DE at the FIG. 172 saine point B. But, since AB, BC, and DE ail lie in the same plane this is impossible, by 7, $ 31.)

·VI, $ 254) PERPENDICULARS AND PARALLELS 221 Р 251. Corollary 2. From a point without a plane, only one line can be druwn perpendicular to the plane. [Hint. If two perpendiculars, as PB and PA, could be drawn from P to the plane MN, then A PBA would contain two right angles so that the sum of the angles of DPBA would be more than two right angles. But this is impossible. Why ?] FIG. 173 M -AB 252. Corollary 3. Through a given point in a straight line, only one plane can be drawn perpendicular to the line. [IIINT. Draw two different perpendiculars in space to the given line at the given point, and apply $8 242, 249. If two such planes exist, their intersections with a plane through the given line violate 7, § 31.] 253. Corollary 4. Through a given point without a straight line, only one plane can be drawn perpendicular to the line. [Hint. Prove by reduction to an absurdity. Show that the intersections of two such perpendicular planes with the plane determined by the given live and given point would violate $ 58.] 254. Corollary 5. All perpendicular lines that can be drawn to a straight line at a given point in it lie in a plane perpendicular to the line at the given point. [IIint. Show that otherwise two perpendicular lines could be drawn to the given line in the same plane at the given point, thus violating 7, $ 31.] EXERCISES 1. Show how to determine a perpendicular to a plane by means of two carpenter's squares. 2. Tell how to test whether or not a flagpole is erect. 3. A spoke of a wheel is perpendicular to the axis on which it turns. Show by $ 254 that it describes a plane in its rotation.

·222 LINES AND PLANES IN SPACE (VI, 8 255 IB M D N 255. Theorem III. Two planes per pendicular to the same line are parallel. [Hint. Show that if the two planes met, say in a point P, $ 253 would be violated.] R FIG. 174 256. Theorem IV. If a plane intersects two parallel planes, the lines of intersection are parallel. Given the plane PQ intersecting the parallel planes MN and RS in AB and CD, respectively. ABCD. [Hint. Prove, by reduction to an absurdity, that AB and CD cannot meet.] To prove M FIG. 175 257. Theorem V. Two lines parallel to a third line (in space) are parallel to each other. Compare $ 50. [Hint. Let BB and CC' be two lines parallel to a third line AA (Fig. 176). The plane determined by CC and the point B on BB' is parallel to AA' ($ 248). Therefore ($ 48) the line of intersection of this plane with the plane of the parallels AA' and BB' is parallel to AA. Hence show, by $ 49, that this ne of intersection coincides with BB', so that BB' and CC' lie in a plane. Finally, show that BB' and CC cannot meet; for, if they did meet, say at a point D, the plane determined by D and AA' would contain (§ 244) both BB' and CC".]

·VI, $ 258) PERPENDICULARS AND PARALLELS 223 258. Theorem VI. If two angles, not in the same plane, have their sides respectively parallel and extending in the same direction, they are equal and their planes are parallel. M B P FIG. 176 Given the angles BAC and B' A'C", lying in the planes MN and PQ, respectively, with AB || A'B', and AC | A'C'. To prove that 2A = 24', and that MNI1 PQ. Proof. Take AB A'B', and AC = A'C". Draw AA', BB, CC", CB, and C'B'. Since AB is equal and parallel to A'B', it follows that ABB' A' is a parallelogramı; Why? hence AA' is equal and parallel to BB'. Why? Similarly, AA' is equal and parallel to CC". Hence BB' is equal and parallel to CC'. $ 257 Then BB'C'C is a parallelogram, and CB= C'B'. Why? Therefore A ABCEA AB'C'. Why? Hence 21=LA. Why? Now PQ) AB. Likewise PQ | AC. $ 248 Therefore, PQI MN for, if not, the line of intersection of PQ and MN would meet either AB or AC (or both extended; hence PQ would not be parallel to each of them. NOTE. The similar theorem for angles that lie in the same plane was proved in $67. As in $ 67, the two angles are supplementary to each other if one pair of corresponding sides extend in opposite directions from the vertices.

·224 LINES AND PLANES IN SPACE (VI, $ 259 F E 259. Theorem VII. A plane perpendicular to one of two parallel lines is perpendicular to the other also. Given the two parallel lines AB and CD, and a plane MN perpendicular to CD at C. To prove that MN is perpendicular to AB. N М. Proof. The parallel lines AB and CD determine a plane ($ 244) FIG. 177 which intersects MN in some line AC. $ 245 Now AC is perpendicular to CD; $ 246 whence AC is perpendicular to AB. $ 60 Draw any line AE in the plane MN through A. Draw CF in MN parallel to AE through C. Then CF is perpendicular to CD. § 246 Hence AE is perpendicular to AB. $ 258 Therefore AB is perpendicular to MN. $ 246 260. Corollary 1. Two lines perpendicular to the same plane ure parallel. [Hint. Let AB and CD (Fig. 177) be perpendicular to the plane MN. Imagine a parallel CD' to AB through C. Then CD' is perpendicular to MN, by $ 259. Hence CD' coincides with CD, by $ 250.] EXERCISES 1. The legs of a table lie along parallel lines in space. What preceding theorem or corollary is illustrated here? Mention other similar illustrations. 2. How many lines can be drawn through a given point parallel to a given plane? If there is more than one such, what is the locus of them all ? 3. Given a plane and two points without it. When will the line through the two points be parallel to the plane ?

·VI, S 261] PERPENDICULARS AND PARALLELS 225 N 261, Theorem VIII. If two straight lines are intersected by three parallel planes, the corresponding segments of these lines are proportional. Given the straight lines AB and CD cut by the parallel planes L, M, and N. Το prove that AE/EB= CF/FT). Proof. Draw BC ineeting the plane M in G. Draw EF, EG, FG, BD, and AC. Then GF || BD, and EG !! AC. $ 256 Now AE/EB=CG/GB, and CF/FI)= CG/GB. $ 145 Therefore AE/EB=CF/FD. Why? Fig. 178 EXERCISES 1. Show that if parallel planes intercept equal segments on one line, they will intercept equal segments on any other line. What is 2. In Fig. 178, AE =5, E B = 4, and CF=6. the value of FD? 3. Two ordinary blocks C and 1) having the respective heights H and h are placed upon each other as shown in the figure. Show that any line AB drawn from the upper surface of C to the lower surface of D will be divided in the ratio H: h by the point P where AB intersects the common surface of the two blocks.

·226 LINES AND PLANES IN SPACE (VI, $ 262 262. Perpendicular Planes. Two planes MN and PQ are said to be perpendicular to each other when any line CD drawn in the one perpendicular to their intersection is perpendicular to the other plane. N FIG. 179. 263. Theorem IX. If a straight line is perpendicular to a plane, every plane containing this line is perpendicular to the given plane. Given the line CD perpendicular to plane MN; and given any plane PQ containing the line CD. To prove that plane PQ is perpendicular to plane MN. Proof. Let AB be the line of intersection of the two planes MN and PQ. Imagine any line C'D' in the plane PQ perpendicular to the line AB. Then CD is parallel to CD. $ 52 But CD is perpendicular to MN; Given hence C'D' is perpendicular to MN. $ 259 Since C'D' is any line of the plane PQ perpendicular to AB, it follows that PQ is perpendicular to MN. $ 262 264. Corollary 1. The line perpendicular to a given plane at a given point lies in any plane through that point perpendicular to the given plane.

·VI, § 265] PERPENDICULARS AND PARALLELS 227 265. Theorem X. If each of two intersecting planes is perpendicular to a third plane, their line of intersection is perpendicular to the third plane. FIG. 180 Given the planes PQ and RS perpendicular to plane MN and intersecting each other in AB. To prove that AB is perpendicular to MN. Proof. Suppose that AB is not perpendicular to My, but that some other line as CF through C, the point coinmon to the three planes, is the perpendicular to MN. Then CF lies in RS and in PQ. $ 264 Hence CF coincides with AB. $ 245 Therefore AB is perpendicular to MN at C. EXERCISES 1. The blades of a side paddle wheel of a steamboat are all perpendicular to the side of the boat. Connect this fact with : one of the preceding theorems. Do the same with the fact that the upright edge of any building is vertical. 2. How many planes can be drawn perpendicular to a given plane and passing through a given line in space ? [HINT. Select a point in the given line, draw the perpendicular line through that point to the given plane, and consider all the planes that can be passed through this perpendicular.]

·228 LINES AND PLANES IN SPACE [VI, & 266 PART III. DIHEDRAL ANGLES 266. Dihedral Angles. The figure formed by two intersecting portions of planes bounded by their line of intersection is called a dihedral angle. The planes forming the dihedral angle are its faces and the line of intersection is its edge. A dihedral angle may be designated by the two letters on its edge, or by the two letters on its edge together with an additional letter on each face. с B E FIG. 181 Thus, in the figure, the planes AD and AB are the faces and AC is the edge of the dihedral angle B-CA-D. The plane angle of a dihedral angle is an angle formed by lines in the two faces perpendicular to the edge at the same point. Thus, GFE is the plane angle of the dihedral angle B-CA-D. The magnitude of a dihedral angle does not depend upon the extent of its faces. If a plane be made to revolve from the position of one face about the edge as an axis to the position of the other face, it turns through the dihedral angle, and the greater the amount of turning, the greater the angle. 267. Measure of Dihedral Angles. The plane angle of a dihedral angle is taken as its measure, so that two dihedral angles are always in the same ratio as the magnitudes of their plane angles. In particular, two dihedral angles are equal when their plane angles are equal.

·VI, § 267] DIHEDRAL ANGLES 229 Dihedral angles are right, acute, obtuse, etc., according as their plane angles are right, acute, obtuse, etc. Similar detinitions exist for complementary dihedral angles, supplementary dihedral angles, vertical dihedral angles, etc. The faces of a right dihedral angle are perpendicular to each other. EXERCISES 1. Read the adjacent dihedral angles in the following figure. Read the vertical, the complementary, the supplementary dihedral angles. (5) 2. If two planes intersect each other, show that the opposite or vertical dihedral angles thus formed are equal. [IIInt. Use § 267.] 3. Show that the dihedral angle through which a door is opened is measured by the plane angle through which the bottom edge of the door moves. 4. Make an instrument for measuring dihedral angles by cutting and folding a piece of heavy paper or cardboard in the manner shown in the figure. 5. What is the number of degrees in one of the dihedral angles of a bay window, it being understood that the bay window consists of three equal upright plane sections, and that their bases for three sides of a regular octagon ?

·230 LINES AND PLANES IN SPACE [VI, 8268 268. Theorem XI. Every point in a plane that bisects a dihedral angle is equidistant from the faces of the angle. (0) R FIG. 182 Given the plane TR bisecting the dihedral angle formed by the planes TM and T'N, so that the dihedral angles M-TS-R and N-TS-R are equal; and given PA and PB perpendicular to TM and TN, respectively, from any point Pin TR. To prove that PA= PB. Proof. Pass a plane through PA and PB and let X be its point of intersection with ST'; let AX, BX, and PX be the intersections of plane PAB with planes TM, TN, and TR. Plane PAB I planes TM and TN. § 263 Then plane PABI ST, $ 265 whence STI AX, BX, and PX. Why? The angles AXP and BXP are the plane angles of the dihedral angles M-ST-R and N-ST-R. Why? Since the dihedral angles are given equal, their plane angles are equal, that is, LAXP= LBXP; whence rt. A AXPrt. A BXP, Why? and therefore PA= PB. Why?

·VI, $ 270] DIHEDRAL ANGLES 231 269. Corollary 1. Any point not in the bisecting plane of a dihedral angle is unequally distant from the two faces. 270. Corollary 2. The plane bisecting a dihedral angle is the locus of all points equally distant from the faces of the ungle. See Nore, $ 99. EXERCISES 1. To what theorem in Plane Geometry does $ 268 correspond? 2. From any point within a dihedral angle perpendiculars are drawn to the faces. Show that the angle formed by these perpendiculars is supplementary to the plane angle of the dihedral angle. 3. Prove that the two adjacent dihedral angles formed by one plane meeting another are supplementary. [IIint. At some point on the edge of the dihedral, erect a plane perpendicular to its edge, and consider the plane angles formed.] 4. What is the locus of all points equidistant from two intersecting planes, each of indefinite extent? 5. What is the locus of all points in space equidistant from two given points ? 6. What is the locus of all points in space equidistant from the circumference of a circle? 7. What is the locus in space of all points equidistant from two intersecting lines? 8. What is the locus of all points equally distant from two parallel lines ? 9. Prove that of the dihedral angles formed by a plane intersecting parallel planes, the alternate and corresponding angles are equal, and the interior angles on the same side of the transversal plane are supplementary. 10. Prove that all plane angles of a dihedral angle are equal.

·232 LINES AND PLANES IN SPACE (VI, $ 271 B PART IV. POLYHEDRAL ANGLES 271. Polyhedral Angles. The figure formel by three or more straight line segments that end in a common point, together with the V-shaped portions of planes determined by pairs of adjacent lines, is called a polyhedral angle. The point at which the lines all meet is called the vertex of the angle. The lines in which the planes meet are its edges; and the V-shaped portions E of the planes between these edges are D its faces. The plane angles in the faces at the FIG. 183. POLYHEDRAL vertex are called the face angles of the ANGLE polyhedral angle. A polyhedral angle is read by naming the vertex and a point in each edge. Thus, in Fig. 183, the polyhedral angle is read V-ABCDE. Two polyhedral angles are congruent if they can be placed so that their vertices coincide and their corresponding edges coincide. A trihedral angle is a polyhedral angle that has three faces. Thus, in Fig. 184, the three planes VAB, VBC, VAC, which meet at V forin the A с trihedral angle V-ABC. Two trihedral angles are congruent if B the three face angles of the one are equal, FIG. 184. TRIHEDRAL ANGLE respectively, to the three face angles of the other, and are arranged in the same order. This can be shown by methods similar to those of $ 45. See also $$ 361,374. If the intersections of a plane with all the faces of a polyhedral angle is a convex polygon, the polyhedral angle is a convex polyhedral angle.

·VI, $ 272] POLYHEDRAL ANGLES 233 272. Theorem XII. The sum of two face angles of a trihedral angle is greater than the third. V A. D B FIG. 185 Given the trihedral angle V-ABC. To prove that LAVB + ZBVC> LAVC. Proof. If ZAVC is equal to or less than either of the other angles, we know the proposition is true without further proof. If LAVC is greater than either of the other angles, lay off any lengths V1 and VC on the sides of Z AVC, and draw AC. Then draw VD in the plane AVC, making 24VD=LAVB. Lay off VB= VD, and draw AB and CB. Then A AVB MAVD. Why? Therefore AB AD, Why? Now AB + BC > AD + DO. Why? Whence, subtracting, BC > DC. Ax. 6 Therefore ZBVC > LDVC. $ 79 By construction LAVB = LAVD. Adding, LAVB+ZBVC> LAVC. EXERCISES 1. If in the trihedral angle V-ABC, LAVB = 60°, and ZBVC 80°, make a statement as to the number of degrees in LAVC. 2. Show that any face angle of a trihedral angle is greater than the difference of the other two.

·234 LINES AND PLANES IN SPACE (VI, S 273 273. Theorem XIII. The sum of the face angles of any convex polyhedral angle is less than four right angles. E B Fig. 186 Given the polyhedral angle V-ABCDE with the edges cut by any plane in the points A, B, C, D, E. To prove that the sum of the face angles of the polyhedral angle is less than four right angles. Proof. Connect any point in the polygon ABCDE with the vertices A, B, C, D, The number of triangles with the common vertex 0 is the same as the number having the vertex V. Now Z VB4+2 VBC > LABO+Z OBC, $ 272 and VAB+ ZVAE>Z BAO + 20AE, etc. Therefore the sum of the base angles of the triangles having V for a common vertex is greater than the sum of the base angles of the triangles having 0 for vertex. But the sum of all the apgles of all the triangles whose vertex is V is equal to the sum of all the angles of all the triangles whose vertex is 0. Why? Therefore the sum of the angles about the vertex V is less than the sum of the angles about 0, that is, less than four right angles.

·VI, $ 273] MISCELLANEOUS EXERCISES 235 MISCELLANEOUS EXERCISES ON CHAPTER VI 1. Lean one book in a slanting position against another book that lies flat on a table, and hold a stretched string parallel to the cover of the slanting book. Can the string have inore than one position? Can the string be horizontal ? Vertical ? 2. Show that if a half-open book is placed on a table, resting on its bottom edges, the back edge of the book is perpendicular to the plane of the table, and the lines of printing are parallel to that plane. 3. Show that the dihedral angle between the pages of an open book is measured by the plane angle between opposite lines of type on the two pages. 4. What is the shape of the end of an ordinary plank after it has been sawed off in a slanting direction, assuming that the opposite faces of the original board are parallel planes ? 5. Prove that the segments of two parallel lines included between parallel planes are equal. [Hint. Pass a plane through the parallel lines and then prove that the givon segments form the opposite sides of a parallelogram. 6. Prove that a plane perpendicular to the edge of a dihe. dral angle is perpendicular to both its faces. [HINT. Use $ 268.] 7. What is the locus of all the points equidistant from the three faces of a trihedral angle? 8. Show that the locus of any given point on a line segment of fixed length, whose ends touch two parallel planes, is a third plane parallel to the given planes. 9. Prove that if three lines are perpendicular to each other at a common point in space, each line is perpendicular to the plane of the other two, and that the planes of the lines (taken in pairs) are perpendicular to each other. Note how this is illustrated on a cube, or in a corner of a room, or in a corner of an ordinary box.

·236 LINES AND PLANES IN SPACE [VI, $ 273 10. The trihedral angle formed when three planes meet each other, so that each is perpendicular to the other two is called a trirectangular trihedral angle. Prove that the edges of a trirectangular trihedral angle are mutually perpendicular by pairs. See $S 246, 265. 11. Prove that the space about a point is divided into eight congruent trirectangular trihedral angles by three planes mutilally perpendicular by pairs at the point. 12. Prove that if a line is parallel to the intersection of two planes, it is parallel to each of the planes. [Fint. Suppose that the line is not parallel to one of the planes and thus argue to an absurdity.] 13. Prove that if a line is parallel to each of two intersecting planes it is parallel to their intersection. 14. Prove that if a line is parallel to a plane, any plano perpendicular to the line is perpendicular to the plane. [Hint. Pass a plane through the given line perpendicular to the given plane and use § 207.] 15. Can a trihedral angle be formed by placing three equilateral triangles so that one vertex of each lies at the vertex of the trihedral angle ? [lint. Use $ 273.] 16. Can a convex polyhedral angle be formed as in Ex. 15 by placing at its vertex one vertex of each of four equilateral triangles? Can this be done with five equilateral triangles ? With six? With more than six ? 17. Can a convex polyhedral angle be formed by placing at its vertex one vertex of each of three squares ? Four squares ? 18. Can a convex polyhedral angle be formed by placing at its vertex one vertex of each of three regular pentagons ? Four ? 19. Show that just five different convex polyhedral angles can be formed as in Exs. 15–18 by placing at a single point one vertex of each of several similar regular polygons.

·VI, $ 273] MISCELLANEOUS EXERCISES 237 B A 20. Show that the sum of the dihedral angles of a trihedral angle lies between two and six right angles. 21. Is there (in general) a point in space that is equidistant from four given points not all of which lie in the same plane ? Give reason for your answer. 22. Given any line l and a plane MN, drop a perpendicular PA from any point N Pin 1 to MN. Prove that I and PA determine a plane perpendicular to MN. M [This plane is called the projecting plane of 1 on MN. Its intersection AB with MN is called the projection of l on. MN. Define similarly the projection of the segment PQ.] 23. Prove that the projection on a plane MN of the line segment joining two points P and Q (Ex. 22) is the line joining the feet A and B of the perpendiculars dropped to the plane from P and Q, respectively. 24. If a line 1 meets a plane MN at N a point B, prove that the projection of l А on MN is the line joining B to the foot M A of a perpendicular let fall from any point Pinl. [The angle ABP between the line l and its projection is called the angle between the line and the plune, or the inclination of the line to the plane.] 25. Prove that the sides of an isosceles triangle make equal angles with any plano containing its base. *26. Show that the length of the projection of any line segment PQ on any plane is the length of PQ times the cosine of the angle between the line and the plane. B

·CHAPTER VII
POLYHEDRONS
CYLINDERS
CONES
PART I.
PRISMS
274. Polyhedrons. A polyhedron is a limited portion of
space completely bounded by planes. The intersections of the
bounding planes are called the edges; the intersections of the
edges, the vertices; and the portions of the bounding planes
bounded by the edges, the faces, of the polyhedron.
IOOSAUEDBON
DODECAHEDBON
OOTAHEDRON
II EXAHEDRON TETRAHEDRON
FIG. 187
A polyhedron of four faces is called a tetrahedron; one of
six faces (for example, a cube), a hexahedron; one of eight
faces, an octahedron; one of twelve faces, a dodecahedron ; one
of twenty faces, an icosahedron.
A diagonal of a polyhedrou is a straight line joining any two
vertices not iħ the same face.
275. Prisms. A prism is a polyhedron, two of whose faces,
called its bases, are congruent polygons in parallel planes, and
whose other faces, called lateral faces, are parallelograms whose
vertices all lie in the bases.
A triangular prism is one whose base is a triangle.
The sum of the areas of the lateral faces of any prism is
called the lateral area of the prism.
238

·VII, S 275] PRISMS 239 The intersections of the lateral faces are the lateral edges of the prisın. The altitude of a prism is the perpendicular distance between its bases. RIGHT PRISMS OBLIQUE PRISMS FIG. 188 E A right prism is one whose lateral edges are perpendicular to its bases. An oblique prism is one whose lateral Al E' edges are oblique to its bases. A regular prism is a right prism whose bases are regular polygons. Any polygon made by a plane which cuts all the lateral edges of a prism, as the polygon A'B'C'D'E' in Fig. 189, called a section of the prism. A right FIG. 189 section is one made by a plane perpendicular to all the lateral edges of the prism, as ABCDE. B D ·240 POLYHEDRONS (VII, $ 276 276. Theorem I. The sections of a prism made by parallel planes cutting all the lateral edges are congruent polygons. N LE' BI E B M Fig. 1.90 Given ABCDE and A'B'C'D'E', sections of the prism MN, made by parallel planes. To prove that ABCDE A'B'C'D'E'. Proof. The sides of the polygon ABCDE are parallel to the sides of the polygon A'B'C'D'E". $ 256 Therefore the polygons are mutnally equilateral. $ 84 Also the polygons are mutually equiangular. $ 258 Therefore polygon ABCDE polygon A'B'C'D'E'. $ 33 EXERCISES 1. How many edges has a tetrahedron ? A hexahedron ? 2. How many diagonals has a hexahedron? A tetrahedron 3. Prove that any two lateral edges of a prism are equal and parallel. 4. Prove that any lateral edge of a right prism is equal to the altitude. 5. Prove that all right sections of a prisin are congruent. 6. Prove that a section of a prism parallel to the base is congruent to the base.

·VII, 8 278] PRISMS 241 277. Theorem II. The lateral area A of a prism is equal to the perimeter pr of a right section multiplied by the lateral edge e; that is, A=p, xe. E D' A B C Fig. 191 Given the prism AD' with a right section PQRST'; let Pr denote the perimeter of the right section, e the lateral edge, and A the lateral area. To prove that A = P, X e. Proof. The lateral area consists of a number of parallelo. grams, each of which has a line equal to AA' for its base. Why? Each of these parallelograms has one of the sides of the right section PQRST for an altitude. Why? Therefore the areas of these parallelograms = the perimeter of PQRST multiplied by AA'. Why? That is A = 1 X e. 278. Corollary 1. The lateral area A of a right prism is equal to the perimeter of its base multiplied by a lateral edge; or A = Pixe, where p denotes the perimeter of the base, and e denotes a lateral edge. EXERCISES 1. Find the altitude of a regular prism, one side of whose triangular base is 5 in. and whose lateral area is 195 sq. in. 2. Show that the lateral area of a regular hexagonal right prism is 4v3.ah, where h is the altitude and a the distanca from the center of the base to one of the sides.

·242 POLYHEDRONS (VII, $ 279 279. Congruent Solids. Any two solids, in particular any two prisms, are said to be congruent when they can be made to coincide completely in all their parts. F I F' G H /E E' A B с B' C' FIG. 192. CONGRUENT PRISMS 280. Theorem III. Two prisms are congruent if three faces that include a trihedral angle of one are congruent respectively, and similarly placed, to three faces that include a trihedral angle of the other. Given the prisms AI and A'I' with face Aface A'J', face AG face A'G', and face AD face A'D'. To prove that prism Al prism A'I'. Proof. & EAF, FAB, and EAB are equal respectively to Á E'AF, FA'B', and E'A'B'. Why? Then trihedral LA=trihedral Z A. $ 271 Place the prism Al on the prism All so that the trihedral LA coincides with its congruent trihedral 24'. Then the face AJ will coincide with the congruent face A'J'; AG with the congruent face A'G'; and AD with A'D'; and points C and D will fall on 0' and D'. $ 33 Since the lateral edges of a prism are parallel and equal, CH coincides with O'H', and DI with D'I'. $$ 257, 49 Therefore the upper bases coincide, and the prisms coincide throughout and are congruent. 281. Corollary 1. Two right prisms having congruent bases and equal altitudes are congruent.

·VII, § 283] PRISMS 243 282. Truncated Prisms. A truncated prism is a portion of a prism included between the base and a section oblique to the base. FIG. 193 (a) FIG. 193 (6) 283. Corollary 2. Two truncated prisms are congruent if three faces including a trihedral angle of the one are congruent respectively to three faces including a trihedral angle of the other. EXERCISES 1. A wooden beam 10 ft. long has a rectangular right cross section whose dimensions are 12 in. by 16 in. If the beam be sawed lengthwise along one of its diagonal planes, show that the resulting triangular prisms are congruent. 2. What will be the lateral area of one of the triangular prisms of Ex. 1 ? Its total area ? Ans. 40 sq. ft.; 414 sq. ft. 3. A carpenter is tp saw from a given square piece of timber a portion of which one end is to be perpendicular to the lateral edges, while three given lateral edges are to be 3 ft. 6 in., 3 ft. 4 in., and 3 ft. long, respectively. Show that these measurements are sufficient to enable him to saw off the desired portion. 4. Show that to make a right prism of any desired shape, it is sufficient to have a pattern of a right section of the desired prism, and the length of one lateral edge. 5. Show that to make a truncated prism of any desired shape, of which one end is a right section, it is sufficient to have a pattern of that end, and the lengths of three given consecutive lateral edges.

·244 POLYHEDRONS (VII, $ 284 284. Theorem IV. An oblique prism is equal in volume to a right prism whose base is a right section of the oblique prism and whose altitude is a lateral edge of the oblique prism. T' S' AA Bl C D' Р A B C FIG. 194 Given the oblique prism AD'; and given a right prism PS' whose base PS is a right section of the prism AD', and whose altitude is equal to a lateral edge 14 of the prism AD'. To prove that prism AD'= prism PS. Proof. The lateral edges of the prison PS' equal the lateral edges of the prism AD'. Const. Therefore AP= A'P', BQ= B'Q', etc. Why? Moreover PQ= P'Q', and the face angles at P, Q, P, Q are right angles. Why? Hence, by superposition, Hace AQ=Face A'Q'. Likewise, Face BRFace B'R', etc. Now, Section PQRST Section P'Q'R'S'T". $ 276 Whence, truncated prism AR truncated prism A'R'. $ 283 Therefore, truncated prism AR + truncated prism PD' = truncated prism A'R' + truncated prism PD'. Ax. 1 It follows that prism AD' = prism PS'.

·VII, § 286] PARALLELEPIPEDS 245 285. Equivalent Solids. Two solids that have the same volume are said to be equivalent, or equal in volume. Thus we proved, in $ 284, that any oblique prism is equivalent to a right prism whose base is a right section of the oblique prism and whose altitude is equal to the lateral edge of the oblique prism. 286. Parallelepipeds. A parallelepiped is a prism whose bases are parallelograins. A right parallelepiped is a parallelepiped whose lateral edges are perpendicular to its bases. RECTANGULAR PARALLELEPIPEI CUBF OBLIQUE PARALLEL EPIPED FIG. 195 A rectangular parallelepiped is a right parallelepiped whose bases are rectangles. A cube is a parallelepiped whose six faces are squares. An oblique parallelepiped is one whose lateral edges are oblique to its bases. EXERCISES 1. Show that the lateral faces of a right parallelepiped are rectangles. 2. Find the sum of all the face angles of a parallelepiped. 3. Find the diagonal of a cube whose edge is 4 in.; 20 in.; A.

·246 POLYHEDRONS [VII, $ 287 287. Theorem V. The plane passed through two diagonally opposite edges of a parallelepiped divides the parallelepiped into two triangular prisms that are equal in volume. Н, D FIG. 196 Given the plane ACGE passing through the opposite edges AE and CG of the parallelepiped AG. To prove that the parallelepiped AG is divided into two equal triangular prisms, ABC-F and ADC-H. Proof. Let IJKL be a right section of the parallelepiped. From the definition of parallelepiped, the opposite faces, AF and DG, and AH and BG, are parallel and congruent. $$ 258,88 Therefore IJ || LK, and IL I JK. $ 256 Then IJKL is a parallelogram. Why? The intersection IK of the right section with the plane ACGE is the diagonal of the O IJKL. Therefore A IJKEA IKL. Why? But the prism ABC-F is equal in volume to the right prism whose base is IJK and altitude AE, and the prism ACD-H is equal in volume to the right prism whose base is ILK and altitude AE. $ 284 But since these right prisms are congruent, $ 281 it follows that prism ABC-F=prism ADC-H. Note. If the faces EFGH and ABCD (Fig. 196) are perpendicular to the edges AE, BF, etc., it is easy to see that the diagonal plane AECG divides the parallelepiped into two congruent triangular prisms. This happens for any rectangular parallelepiped. + ·VII, § 287) PARALLELEPIPEDS 247 EXERCISES 1. Prove that the diagonals of a rectangular parallelepiped are equal, and that the square of the diagonal is equal to the suin of the squares of the three edges that meet in any Vertex. [Hint. The diagonal is the hypotenuse of a right triangle whose sides are one of the edges and a diagonal of one face; the diagonal of the face is the hypotenuse of a right triangle whose sides are two of the edges.] 2. Find the length of the diagonal of a rectangular parallelepiped whose edges are 8, 10, 12. 3. Find the edge of a cube whose diagonal is 64 in. 4. Prove that the diagonals of a parallelepiped bisect each other. [Hint. Consider each pair of diagonals separately, and apply $ 87 to the diagonal plane in which they lie.] 5. Prove that if the right section IJKL (Fig. 196) of a parallelepiped is a rectangle, the two diagonal planes ACGE and BDIIF divide the parallelepiped into four triangular prisms that are equal in volune. 6. A tank in the form of a rectangular parallelepiped that holds 100 gal. is divided into four compartinents by two vertical diagonal planes. What is the capacity of each compartment ? 7. A cube each of whose edges is 1 ft. long is called a unit cube; its volume is one cubic foot. If six such cubes are placed side by side in two rows of three each, they form a rectangular parallelepiped 2 ft. wide, 1 ft. high, and 3 ft. long. What is the volume of this parallelepiped? What is the volume of each of the triangular prisms into which it is divided by a diagonal plane ? 8. How many unit cubes are there in a cube each of whose edges is 5 units long? 9. How many unit cubes are there in a rectangular parallelepiped 3 units long, 4 units wide, and 2 units high? What is the volume of this parallelepiped ?

·248 POLYHEDRONS (VIT, 8 288 288. Volume of a Rectangular Parallelepiped. The three edges of a rectangular parallelepiped which meet at a common point are called its dimensions. In Chapter IV ($ 181), we assumed (without proof) the well-known principle that the area of a rectangle is equal to the product of its two dimensions. Simi. larly, we shall now assume that the vol- FIG. 197 ume of a rectangular parallelepiped is equal to the product of its three dimensions, that is, to the product of its length, breadth, and height; i.e. For any rectangular parallelepiped the volume V is v=lxbxh, where l, b, h denote the length, breadth, and height of the parallelepiped. B The student is reminded that the meaning of the principle is that, if the three dimensions are each measured in terms of the same unit of length, then the volume in terms of the corresponding unit cube is the product of the three A D dimensions. Fig. 198 The following corollaries result at once from this principle: 289. Corollary 1. Two rectangular parallelepipeds having congruent bases are to each other as their altitudes. [Hint. If 1, b, and h represent the dimensions of the one parallelehl piped, then 1, 6, and h' will represent h the dimensions of the other. The corresponding volumes will therefore be to each other in the ratio (15h)/(lbh'), that is, in the ratio h/h!.] FIG. 199 N ·VII, § 292] PARALLELEPIPEDS i 290. Corollary 2. Two rectangular parallelepipeds having equal altitudes are to each other as their bases. 291. Corollary 3. The volume of a cube is equal to the cube of its edge. 292. Corollary 4. The volume V of any rectangular parallelepiped is the product of the areu of its base B and its altitude h; that is, V=Bxh. EXERCISES 1. Two rectangular parallelepipeds with equal altitudes have bases containing 10 sq. in. and 15 sq. in., respectively. The volume of the first is 56 cu. ft. Find the volume of the second. Ans. 84 cu. ft. 2. Compare the volume of the rectangular parallelepiped whose dimensions are 8 in., 10 in., 11 in. with the one whose dimensions are 1 ft., 1 ft., and 16 in. 3. In a lot 120 ft. long and 66 ft. wide a cellar is to be dug for a building. The cellar is to be 44 ft. long, 36 ft. wide, and 7 ft. deep. The earth removed is to be used to fill the surrounding yard. What depth of fill can be made ? 4. A standard (U.S.) gallon contains 231 cu. in. gallons can be put in a tank of the form of a rectangular parallelepiped that is 2 ft. high, 1} ft. wide, and 3 ft long? 5. How many gallons (see Ex. 4) are there in 1 cu. ft. ? 6. Find the size of a cubical tank that will contain 50 gal. 7. Find the edge of a cube whose volume is 1728 cu. in.; of a cube whose volume is 1500 cu. in. 8. Find the diagonal of a cube whose volume is 521 cu. in. 9. If the volume of one cube is twice that of another, how do their edges compare ? Ans. V2:1. 10. Find the edge of a cube whose total surface is 60 sq. ft. 11. The edge of a cube is al. Find the area of a section made by a plane through two diagonally opposite edges. How many ·250 POLYHEDRONS (VII, § 293 293. Theorem VI. The volume V of any parallelepiped is equal to the product of its base B and its altitude h; that is, V=B xh. II I M III N ih HB h B!! Fig. 200 Given the parallelepiped I with its volume denoted by V, its base by B, and its altitude by h. To prove that V-B X h. Proof. Produce AC and all the other edges of 1 that are parallel to 4C. On the prolongation of AC take DE = AC, and through D and E pass planes perpendicular to AE, forming the right parallelepiped II whose base is B'. Then I=II. § 284 Prolong FE and all the other edges of II that are parallel to FE. On the prolongation of FE take MN= FE, and through M and N pass planes perpendicular to FN, forming the rectangular parallelepiped III whose base is B". Then II = III. Why? Therefore I III. Why? Moreover B: Why? and h, the altitude of I, is equal to the altitude of III. Why? But the volume of III is B'' x h, by $ 288; hence the volume of I is V=B" xh = BX h. B";

·VII, § 296]
VOLUME FORMULAS
251
294. Theorem VII. The volume V of any triangular prism
is equal to the product of its base B and its altitude h; that
is, V=B xh.
N'
pl
MP
N
B
M
FIG. 201
Given the triangular prism LMN-N' whose base B is the
triangle LMN, and whose altitude is .
To prove that the voluine V of MN-N' B xh.
Proof. Complete the parallelepiped LMPN-P'.
[The remainder of the proof is left to the student. Use § 293.]
295. Corollary 1. The volume V of any prism is equal to
the product of its base B and its altitude h; that is, V=BX1.
FIG. 202
[Hint. Any prism may be divided into triangular prisms by diagonal
planes.]
296. Corollary 2. Prisms having equivalent bases and equal
altitudes are equal.

·252 POLYHEDRONS [VII, $ 296 EXERCISES 1. Describe one or more ways in which a parallelepiped may be distorted and yet have its volume reinain unchanged. 2. The base of a parallelepiped is a rhombus one of whose diagonals is equal to its side. The altitude of the parallelepiped is a, and is also equal to a side of the base. Find the volume of the parallelepiped. Ans. a V3/2. 3. The altitude of a parallelepiped is 3 in., and a diagonal of a base divides the base into two equilateral triangles, cach side of which is 6 in. Find the volume of the parallelepiped. 4. The volume of a rectangular parallelepiped is 2430 cu. in. and its edges are in the ratio of 3, 5, and 6. Find its edges. 5. The altitude of a prism is 6 in. and its base is a square each side of which is 3 in. Find its volume. 6. Show that two prisms with equal bases are to each other as their altitudes; and that those with eqnal altitudes are to each other as their bases. 7. A clay cube having a 2-in. edge is molded into the forin of a triangular prism of height 3 in. What is the area of its base? Does it make a difference in the answer whether the prism is made right or oblique ? Explain. 8. Assuming that iron weighs about 150 lb. per cu. ft., find the weight of a rod 3 ft. long, whose cross section is a rectangle 11 by 2 in. 9. With the data of Ex. 8, find the weight of an iron rod 2 ft. 6 in. long, whose cross section is a regular hexagon 1 in. on each side. 10. What must be the length of the side of an equilateral triangle in order that a triangular prism erected upon it and of height 1 ft. shall have a volume of 1 cu. ft. ? Solve the same problem, when it is assumed that the base is a regular hexagon.

·VII, § 297] PYRAMIDS 253 PART II. PYRAMIDS 297. Pyramids. A pyramid is a polyhedron bounded by a polygon, called its base, and several triangles that have a common vertex. The triangles are called the lateral faces, the common vertex is called the vertex of the pyramid, and the perpendicular distance from the vertex to the base is called the altitude. E P с A B FIG. 203. PYRAMID A pyramid is triangular, quadrangular, etc., according as its base is a triangle, a quadrilateral, etc. A regular pyramid is one whose base is a regular polygon and whose vertex lies in the perpendicular erected at the center of the base. A A D B C FIG. 201. REGULAR PYRAMID The slant height of a regular pyramid is the altitude of any one of its triangular faces (VII in Fig. 204).

·254 POLYHEDRONS (VII, § 297 A truncated pyramid is the portion of a pyramid included between the base and any section made by a plane cutting all the lateral edges. A frustum of a pyramid is a portion included between the base and a section made by a plane parallel to the base. FIG. 205. FRUSTUM OF A PYRAMID The altitude of a frustum is the length of the perpendicular between the planes of its bases. The lateral faces of a frustum of a regular pyramid are congruent trapezoids. The slant height of the frustum of a regular pyramirl, is the altitude of one of the trapezoids forming its faces. EXERCISES 1. Of which type are the celebrated Egyptian pyramids ? 2. Prove the equality of the lateral edges of a regular pyramid. Of those of a frustum of a regular pyramid. 3. Prove that the faces of any frustum of a pyramid are trapezoids. 4. Prove the statement made in $ 297 that the faces of a frustum of a regular pyramid are congruent trapezoids. 5. Prove that the lateral faces of a regular pyramid are congruent isosceles triangles; hence show that the slant height, measured on any face, is the same as that measured on any other face. 6. Prove that any triangular pyramid is a tetrahedron. State and prove the converse.

·VII, $ 299) PYRAMIDS 255 298. Theorem VIII. The lateral area A of a regular pyramid is equal to one half the product of the perimeter of its base p, and its slant height l; that is, A=px1/2. A B FIG. 206 Given the regular pyramid V-ABCDE with the slant height 1, the lateral area A, and the perimeter of the base p. To prove that A=px1/2. Proof. A the sum of the areas of the triangles VAB, VBC, utc. Hence A =[AB + BC+...] x 1/2=p x1/2. Why? 299. Corollary 1. The lateral area of the frustum of a regular pyramil is equal to one half the product of the sum of the perimeters of the bases and its slant height. (See $ 191.] EXERCISES 1. The slant height of a regular hexagonal pyramid is 10 ft. Each side of its base is 8 ft. What is its lateral area ? Also, what is its total area? Ans. 240 sq. ft. ; 406.27 sq. ft. 2. The altitude of a regular quadrangular pyramid is 4 in. One side of its base is 6 in. What is its lateral area ? What is its total area ? Ans. 60 sq. in.; 96 sq. in. 3. Find the lateral area of the frustum formed by a plane bisecting the altitude of the pyramid mentioned in Ex. 2 Find its total area.

·256 POLYHEDRONS [VII, $ 300 300. Theorem IX. If a pyramid is cut by a plane parallel to the base, (a) The altitude and the lateral edges are divided proportionally; (6) The section is a polygon similar to the base. B' A "S'D B C FIG. 207 حسنة Given the pyramid V-ABCDE cut by a plane A'D' parallel to the base AD. To prove (a) that VS/VS' = VA/VA' = VB/VB' = ..., etc, (6) that the section A'B'C'D'E' is similar to the base. Proof of (a) Pass a plane through V parallel to the base and apply $ 261. Proof of (b) AVAB ~ A VA'B'; A VBCA VB'C', etc. Why? Therefore AB A'B' =VB/VB'; VB/VB' = BC/B'C', etc.; Why? and hence AB/ A'B' = BC/B'C' = CD/CD' Ax. 9 Thus, the polygons ABCDE and A'B'C'D' E' have their corresponding sides proportional. Moreover, the same polygons are inutually equiangular. $ 258 Hence ABCDE - A'B'C'D' E'. $ 165 ·VII, § 302] PYRAMIDS 257 Ty 301. Corollary 1. Parallel sections of a pyramid are to each other as the squares of their distances from the vertex. Proof: In Fig. 207, ABCDE/A'B'C'D' E' = AB' A'B”. SS 195, 300 But : AB/A' B' = VB/VB', Why? and also VS/VS! VB/VB'; (a), $ 300 hence AB A'B' = VS/VS. Ax. 9 Whence, squaring, AB'/ A' B^2 =VS/V82. Hence ABCDE/A'B'C'D'E' V$€/TS" 302. Corollary 2. If two pyramids that have equivalent bases and equal altitudes are cut by planes parallel to their bases and at equal distances from their vertices, the sections are equivalent. [Hint. Represent by "R and R' the areas of the two sections, and by B and Bl the areas of the bases. Let h be the common altitude of the pyramids, and k the distance from the vertex of either pyramid to the section made in it. Then R/B = k2/h2 and R'/B! = k/12 ($ 301); hence R/B - R/B'. But B = B' by hypothesis ; hence R = R'.] EXERCISES 1. Compare the areas of two sections of a pyramid whose perpendicular distances from the vertex are 3 in. and 4 in. respectively. Does it make any difference in your answer whether the pyramid is of one shape or another? Ans. 9:16. 2. The altitude of a pyramid with a square base is 16 in., the area of a section parallel to the base and 10 in. from the vertex is 564 sq. in. Find the area of the base. 3. The bases of the frustum of a regular pyramid are equilateral triangles whose sides are 10 in. and 18 in, respectively; the altitude of the frustuin is & in. Find the altitude of the pyramid of which the given figure is a frustun. Ans. 18 in. 4. The altitude of a pyramid is H. At what distance from the vertex must a plane be passed parallel to the base so that the section formed shall be (1) one half as large as the base ? (2) one third ? (3) one ninth ?

·258 POLYHEDRONS (VII, $ 303 303. Theorem X. Two triangular pyramids having equivalent bases and equal altitudes are equivalent. V! H А с AP B FIG. 208 To prove Given the pyramids V-ABC and V'-A'B'C' having equivalent bases ABC, A'B'C', and a common altitude AH. V-ABC=V-A'B'C". Proof. The proof of the theorem consists in showing that the pyramids V-ABC and V'-A'B'C' cannot differ in volume by as much as any given amount, however small. This means that the two volumes are actually equal, for if they were unequal, they would differ by as much as some fixed amount, - in fact, that is what unequal means. We proceed, then, to show that V-ABC and V'-A'B'C' cânnot differ by as much as any given amount, however small. Divide the altitude AH into a number of equal parts, and through the points of division pass planes parallel to the plane of the bases. The corresponding sections made by any one of these planes in the two pyramids are equivalent. Now inscribe in each pyramid a series of prisms baving the triangular sections as upper bases and the distance between the sections as their common altitude. $ 302 ·VII, S 304] PYRAMIDS 259 Each pair of corresponding prisms in the two pyramids are then equivalent. $ 296 Therefore, the sum of the prisms inscribed in V-ABC is equivalent to the sum of the prisms inscribed in V-A'B'C". If the number of divisions into which Al is divided is taken sufficiently large, the sum of the prisms in V-ABC may be made to differ froin the volume of V-ABC by less than any given amount. Likewise, by taking the number of divisions in AH sufficiently large, the sum of the prisms in V'-A'B'C may be made to differ from the volume of V'-A'B'C' by less than the same given amount, however small it has been taken. Since, by taking the number of divisions in AH sufficiently large, the voluines of V-ABC and V'-A'B'C" differ by less than any given amount from these equal sums, the pyramids must differ from each other by less than the same given amount, and this is what we were to show. Compare $ 211. 304. Corollary 1. Any two pyramids having equivalent bases and equal altitudes are equivalent. FIG. 209 (Hint. Divide each pyramid into triangular pyramids.] EXERCISES 1. What is the locus of the vertices of all pyramids whose bases and volumes are the same ? 2. Prove that if the base of a pyramid is a parallelogram, the plane determined by its vertex and either diagonal of its base divides it into two equivalent triangular pyramids.

·260 POLYHEDRONS (VII, § 305 305. Theorem XI. The volume V of a triangular pyramid is equal to one third the product of its base B, and its altitude h; that is, V=BX1/3. 요 M Fig. 210 Given the triangular pyramid 0-LMN. To prove that the volume V of 0-LMN equals its base B times its altitude h; that is, V=BX 1/3. Proof. Construct a triangular prism MP having LMN for its base, and its lateral edges equal and parallel to the edge OM. The prism MP is made up of the triangular pyramid 0-LMN and the quadrangular pyramid 0-LNPQ. Pass a plane through OQ and ON dividing the quadrangular pyramid into two triangular pyramids, 0-LNQ and 0-NQP Pyramid 0-LNQ = pyramid 0-NQP. § 303 Pyramid 0-NQP may be read N-QOP. Pyramid N-QOP= pyramid 0-LMN. $ 303 Therefore, the three triangular pyramids are equal and O-LMN is one third the prism. But the volume of the prism is equal to the product of its base and its altitude. $ 294 Therefore, pyramid 0-LMN = f the product of its base and its altitude.

·VII, § 307] PYRAMIDS 261 306. Corollary 1. The volume V of any pyramid is equal to one third the product of its base B and its altitude h; that is, V = Bh/3. Fig. 211 [Hint. Divide the pyramid into triangular pyramids and apply $ 305.] 307. Theorem XII. If a frustum of a pyramid has bases B and B' and altitude h, and is cut from a pyramid P whose base is B and whose altitude is II, the volume V of the frustum is given by the formula : ВНІ B'(H --N) V 3 3 Given the pyramid P with base B and altitude II, and a frustum of it with bases B and B' and altitude h. To prove that the volune V of the frus- L N tum is B BH B'(HT – ?). 3 3 FIG. 212 Proof. The frustum is the difference between the two pyramids P and P', where P' has the base B" and the same vertex as P. The volume of P is BH/3. Why? Since the altitude of P' is II - h, its volume is B'(H — ). Why? 3 Hence V=P-P BH_ B'(II -"). 3 3 M ·262 POLYHEDRONS (VII, 8 308 308. Corollary 1. The volume V of a frustum of a puramid of bases B and B'* and altitude h is given by the formula: V=(B + B' +V BB') /3. Outline of Proof. By $ 301, B'/B=(H-1)/H, P using the notation of $ 307. Hence VB/VB=(H –n)/H=1-h/H, whence H=hVB/VBVB'). But, by $ 307, N V= BH/3 - B'(H - h)/3 B (B - B')H/3 + B'h/3. Substituting the value of HI just found, we M have FIG. 213 V (B - B') VB [.B + V BB! + B']h/3. VB-VB + B B']n/3 EXERCISES 1. Find the altitude of a triangular pyramid whose volume is 50 cu. in, and whose base is 12 sq. in. Ans. 121 in. 2. If a prism and a pyramid have a common base and alti. tude, what is the ratio of their volumes ? 3. If the base of a pyramid is a square and its altitude is 3 ft., how long must each side of the square be in order that the volume may be 16 cu. ft.? 4. Show that the volume of the tetrahedron, all of whose edges are equal to a, is V2 a3/12. [HINT. See Th. XXXIII, § 102.] 5. Find the volume of a frustum of the pyramid of Ex. 1 cut off by a plane 6 in. from the base. 6. Find the volume of each of the parts into which the pyramid of Ex. 3 is cut by two planes parallel to its base which trisect the altitude. 1 ·VII, $ 309] CYLINDERS 263 PART III. CYLINDERS AND CONES 309. Cylinders. A cylindrical surface is a curved surface generated by a moving straight line, called the generatrix, which moves always parallel to itself and constantly passes through a fixed curve called the directrix. The generatrix in any one position is called an element of the surface. One element and only one can be drawn through a given point on the cylindrical surface. A cylinder is a solid bounded by a cylindrical surface and two parallel planes. The two plane surfaces are called the bases, and the cylindrical surface is called the lateral surface. The altitude of a cylinder is the length of the perpendicular between the bases. A right section of a cylinder is a section made by a plane perpendicular to all its elemnents. CYLINDRICAL SURFACE OBLIQUE CYLINDER RIGHT CYLINDER FIG. 214 A circular cylinder is one whose bases are circles. A right cylinder is one whose elements are all perpendicular to its bases; otherwise, the cylinder is said to be oblique. A right circular cylinder is a right cylinder whose base is 'a circle. Such a cylinder can be generated by the revolution of ·264 CYLINDERS (VII, § 310 a rectangle about one of its sides as an axis; for this reason a right circular cylinder is sometimes called a cylinder of revolution. 310. Postulate. A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder. FIG. 215 In order to study the properties of the cylinder the following postulate is needed: A cylinder and a prism inscribed within it may be made to differ by as little as we please, both in lateral area and in volume, by making the number of sides of the base of the prism sufficiently great, while the length of each of those sides becomes sufficiently small. The length of an edge of the inscribed prism is equal to the length of an element of the cylinder (see Ex. 5, p. 235); and, by increasing the number of sides of the inscribed prism, the base of the prism approaches, both in area and in the length of its perimeter, as nearly as we please to the base of the cylinder. This latter fact we assume, as in $ 210. We shall now proceed to show that the theorems already proved for prisms can be extended to cylinders by the use of the preceding postulate.

·VII, § 313] AREAS AND VOLUMES 265 311. Theorem XIII. The lateral area A of any cylinder is equal to the product of an element l and the perimeter p of a right section ; that is, A=lxp. n Fig. 216 Outline of Proof. In order to prove the theorem, inscribe in the cylinder any prism whose base is a polygon of n sides. Then, for this prism, by $ 277: (1) A'=l xp', where A' and p' are, respectively, the lateral area and the perimeter of the right section of the prism; and where l is the length of an edge of the prism, which is equal to an element of the cylinder. As the number n of sides increases so that the length of each side approaches zero, A comes to differ from A by as little as we please; $ 310 lxp comes to differ from lxp by as little as we please. $ 310. Hence, by (1), A comes to differ from l xp by as little as we please. It follows, as in $ 303, that A=Ix p. 312. Corollary 1. The lateral area of a right circular cylinder is equal to 2 arh, where r is the radius of the circular base and h is the altitude of the cylinder. See § 214. 313. Corollary 2. The lateral area of any cylinder whose right section is a circle is equal to 2 mrl, where r is the radius of the right section, and I is the length of an element.

·266 CYLINDERS (VII, § 314 314. Theorem XIV. The volume v of any cylinder is equal to the product of its base B and its altitude 'h; that is, V=Bxh. [The proof is left to the student. Inscribe a prism of n sides in the cylinder and use § 205 and § 310. Follow the steps suggested by $ 311.] 315. Corollary 1. The volume of a circular cylinder is equal to wh, where r is the radius of the base and h is the altitude of the cylinder. See $ 216. EXERCISES [In these exercises, use the approximate value = = 22/7.] 1. In a steam engine 65 flues, or cylindrical pipes, each 2 in. in outside diameter and 12 ft. long, convey the heat from the fire-box through to the water. How much heating surface is presented to the water ? Ans. 4084 sq. ft. 2. Neglecting the lap, how much tin is required to make a stovepipe 10 ft. long and 8 in. in diameter? 3. A right circular cylinder has the radius of its base equal to 3 in. How great must its altitude be in order that it shall have a lateral area of 30 sq. in. ? 4. Find the total area, including the ends, of a covered tin can whose diameter is 4 in. and whose height is 6 in. Ans. 1000 sq. in. 5. Derive a general formula for the total area (including the bases) of a right circular cylinder whose height is h and the radius of whose base is r. 6. What fraction of the metal in a tin can 5 in. wide and 5 in. high is used to make the top and bottom? What to make the circular sides? 7. If the diameter of a well is 7 ft. and the water is 10 ft. deep, how many gallons of water are there in it, reckoning 7 gal. to the cubic foot? Ans. 2887.5 gals.

·VII, § 315] AREAS AND VOLUMES 267 8. When a body is placed in a cylindrical tumbler of water 3 in. in diameter, the water level rises 1 in. What is the volume of the body? Note that a method for finding the volume of a body of any shape is here illustrated. 9. Show that the volume V and the lateral area A of a right circular cylinder are connected by the relation V= AXT/2. 10. One gallon is 231 cubic inches. At what heights on a cylindrical measuring can whose base is 6 in. in diameter will the marks for 1 gallon, 1 quart, 2 quarts, 3 quarts, be made ? 11. Find the total area of the gallon measuring can of Ex. 10. 12. Having given the total surface T of a right circular cylinder, in which the height is equal to the diameter of the base, find the volume V. [Hint. - Call the height h.] 13. Find the volume of metal per foot of length in a pipe whose outer diameter is 3 in., and whose inner diameter is 3 in. Hence find the weight per foot of length if the pipe is iron, it being given that iron weighs (about) 450 lb. per cubic foot. 14. If the outer and inner diameters of a tube are D and d, respectively, show that the volume in a length l is al(D2 - d^)/4. If the thickness of the tube is denoted by t, show that t= (D-d)/2, and hence that the volume in a length l is All(D+d)/2 15. Show that the volume of the tube of Ex. 14 can also be represented by the formula ald+t); or by the formula alt(D-t), in the notation of Ex. 14. 16. What is the volume of the largest beam of square cross sectiou that can be cut froin a circular log 16 in. in diameter and 10 ft. long? What fraction of the log is wasted ?

·268 CONES (VII, $ 316 316. Cones. A conical surface is a surface generated by a moving straight line AB, called the generatrix, which passes always through a fixed point V, called the vertex, and constantly FIG. 217 lower nappes. intersects a fixed curve, called the directrix. A conical surface thus consists of two parts, which are called the upper and The generatrix in any one position is called an element of the surface. A cone is a solid bounded by a conical surface and a plane which cuts all its elements. The plane is then called the base of the cone; and the conical surface is called the lateral surface of the cone. The altitude of a cone is the perpendicular distance from its vertex to its base. --- Fig. 218 A circular cone is one whose base is a circle. The axis of a circular cone is the line joining the vertex to the center of the base.

·VII, S 3161 AREAS AND VOLUMES 269 A right circular cone is a circular cone whose axis is perpendicular to the base. Such a cone is also called a cone of revolution, since it may be generated by revolving a right triangle about one of its sides as an axis. O height slant th R L B M Fig. 219. RIGHT CIRCULAR CONE AND SECTION PARALLEL TO BASE The slant height of a cone of revolution is the length of one of its elements. A frustum of a cone is the portion of a cone included between the base and a section parallel to the base and cutting all the elements. B' R B M Fig. 220. FRUSTUM OF A CONE The lateral surface of a frustum of a cone is the portion of the lateral surface of the cone included between the bases of the frustum. The slant height of a frustum of a cone of revolution is the portion of any element of the cone included between the bases.

·cones of 270 CONES [VII, 8 317 317. Postulate. A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is inscribed in the base of the cone. The following postulate, corresponding to that of $ 310, is needed in the study of the cone : A cone and pyramid inscribed within it may be made to differ by as little as we please, both in lateral area and in volume, by making the number of sides of the pyramid sufficiently great, while the length of each side of the base becomes sufficiently small. FIG. 221. CONE WITH INSCRIBED PYRAMID The base of the inscribed pyramid approaches, both in area and in perimeter, the base of the cone ($ 310); and the altitude of any face of the pyramid approaches an element of the cone, as the pyramid approaches the cone. If the cone is a right circular cone, the pyramid can be made a regular pyramid; then the slant height of the pyramid approaches the slant height of the cone. 318. Restriction. The word cone as used hereafter in this book will be understood to refer to a circular cone only. The preceding postulate applies, however, to any kind of cone; and it may be used to obtain results for any form in the manner illustrated below for circular cones only. We proceed to extend to circular cones certain theorems already proved for pyramids.

·VII, § 321) AREAS AND VOLUMES 271 319. Theorem XV. The lateral area A of a right circular cone is equal to one half the product of its slant height l and the circumference p of its base; that is, A=1x p/2. Outline of Proof. Inscribe a regular pyramid of n faces in the cone (see Fig. 221); then, by $ 298, the lateral area A of the pyramid is equal to one half the product of its slant height l' and the perimeter p' of its base; that is, A = xp'; and this formula is correct no matter how large n may be. By taking n sufficiently large, A' comes to differ by as little as we please from A; while l' and p' come to differ by as little as we please from 1 and p, respectively. $317. Whence, as in $ 311, A= tlx p. 320. Corollary 1. The lateral area of a right circular cone is mr.l, where r is the radius of the base and l is the slant height. See $$ 319 and 214. 321. Corollary 2. The lateral area of a frustum of a right circular cone is equal to one half the product of its slant height and the sum of the circumferences of its bases. B' P R LY B м. Fig. 222 [The proof is left to the student. Inscribe a frustum of a regular pyramid in the given frustum of a cone, and use $ 317 and $ 299.1.

·272 CONES [VII, $ 322 322. Theorem XVI. Any section of a circular cone parallel to its base is a circle whose area is to that of the base as the square of its distance from the vertex is to the square of the altitude of the cone. [Hint. To prove that the section is a circle, pass any two planes through the axis of the cylinder, and show that their intersections with the section are equal. Then inscribe a regular pyramid and proceed as in § 319, using $S 301 and 317.] 323. Theorem XVII. The volume V of a cone is equal to one third the product of its base B and its altitude h; that is, V = Bh/3. [HINT. Use $$ 317, 306, and proceed as in § 319.] 324. Corollary 1. If from any cone whose base is B and whose altitude is H, a frustum is cut, whose upper base is B' and whose altitude is hi, the volume V of the frustum is BH B' (1-1) V 3 3 height H BA slant th ER B M FIG. 223 325. Corollary 2. The volume of a frustum of any cone is equal to the sum of three cones whose common altitude is the altitude of the frustum and whose bases are the two bases and a mean proportional between them. THINT. Use 88 322, 324, noting also 308.]

·VII, 325] AREAS AND VOLUMES 273 EXERCISES 1. The altitude of a right circular cone is 12 in. and the radius of the base 9 in. Find the lateral area and the total area of the cone. Ans. 4244 sq. in.; 6789 sq. in. 2. How many square yards of canvas are there in a conical tent 12 ft. in diameter and 8 ft. high ? 3. The total area of a right circular cone whose altitude is 10 in. is 280 sq. in. Find the total area of the cone cut off by a plane parallel to the base and 6 in. from the vertex. 4. The altitude of a right circular cone is H. How far from the vertex must a plane be passed parallel to the base so that the lateral area of the cone cut off shall be one half that of the original cone? Ans. H/V2. [Hint. First prove that the area of the cross section made by the plane will be one half the area of the base. Then apply $ 322.] 5. The slant height and the diameter of the base of a right circular cone are each equal to l. Find the total area, including the base. 6. The circumference of the base of a circular cone is 11 ft. and its height is 8 ft. What is its volume ? 242 Ans. or 25 cu. ft. 3п 7. If the height of a circular cone is 10 ft., what must be the radius of its base in order that the volume may be 30 cu. ft. ? 8. A frustum of a cone is 1 ft. high and the radii of its bases are respectively 9 ft. and 4 ft. Find its volume. B' 9. If yo and R are the radii of the bases of the frustum of a cone and l is R its slant height, find the formula for its L B M volume. > 2 T ·274 POLYHEDRONS (VII, $ 326 PART IV. GENERAL THEOREMS ON POLYHEDRONS SIMILARITY REGULAR SOLIDS VOLUMES 326. Theorem XVIII. Two triangular pyramids that have trihedral angle of the one equal to a trihedral angle of the other are to each other as the products of the edges including the equal trihedral angles. H G' K G K F' F' FIG. 224 Given the triangular pyramids ()-FGH and O-F'G'HI', with the trihedral 20= trihedral 20', and with volumes denoted by V and V', respectively. To prove that V/V =[OF.OG. OH]/[O'F". O'G'. O'H']. Proof. Place pyramid 0-F"G'H' so that trihedral 20' will coincide with trihedral 20). From II and Ill draw IIK and I'K' perpendicular to the plane OFG. Then V/V =[A OFG. HK]/[A OF'G'. II'K'] =[A OFG]/[A OF"G'] : [UK/II'K']. Why? AOFG OF. OG But § 193 A OFG' OF. 00 Again, let the plane determined by HK and H'K' intersect plane OFG in line OK'K. Then rt. A OKH - rt. A OK'H'. Why? Therefore HK/H'K'=0H/OH Why? V OF. OG OH Therefore OF.OG. OH V OF. OG ОН! O'F.O'G'. O'H ·VII, $ 330] GENERAL THEOREMS 275 327. Corollary 1. Two triangular prisms that have a trihedral angle of the one equal to a trihedral (ingle of the other are to each other as the products of the edges including the trihedral angles. [Hint. Break the prism up into triangular pyramids, and use $ 326 and Thoorem II, § 144.]. 328. Corollary 2. Two parallelepipeds that have a trihedral angle of the one equal to a trihedral angle of the other are to each other as the products of the edges including the trihedral angle. 329. Similar Tetrahedrons. Two tetrahedrons (that is, triangular pyramids) are said to be similar if their faces are similar each to each and similarly placed. 330. Theorem XIX. The volumes of two similar tetrahedrons are to each other as the cubes of any two corresponding edges. H' G GP F F FIG. 225 $ 329 Given the similar tetrahedrons 0-FGH and 0-F'G'H' with the volumes denoted by V and V', and with OF and OF" two corresponding edges. To prove that V/V'=OF/ÖF". SS 158, 271 Proof. Trihedral 20=trihedral Z0'. Therefore OF. OG . OII OF OG OH V/V': § 326 O'F. O'G'. O'II' O'F' O'G' O'H' But OF/O'F' = OG/O'G' = OH/O'H'. Why? Therefore V/V =(OF/O'F")(OF/O'F') (OF/O'F")= OF/O'F".

·276 POLYHEDRONS [VII, § 331 331. Similar Polyhedrons. In general, similar polyhedrons are polyhedrons which have the same number of faces similar each to each and similarly placed, and their corresponding polyhedral angles equal. In the case of similar tetrahedrons, the trihedral angles of the one are necessarily equal to those of the other, if we know only that the faces are similar each to each, since the similarity of the faces makes the three face angles at each vertex equal in the two tetrahedrons, by $ 158. By $ 330 and Theorem H, § 144, we can show that any two similar polyhedrons are to each other as the cubes of any two corresponding edges. 332. The Regular Solids. A regular polyhedron is one whose faces are all congruent regular polygons and whose polyhedral angles are all likewise congruent. Five types of such polyhedrons are represented below. ICOS AHEDRON DODECAHEDRON OOTAHEDRON CUBE TETRAUEDBON FIG. 226. THE FIVE REGULAR SOLIDS We shall now show that the above types are the only possible types of regular polyhedrons.

·VII, § 333) GENERAL THEOREMS 277 There exist only five different types of 333. Theorem XX. regular polyhedrons. Proof. The proof is based upon two facts: (1) that any polyhedral angle has at least three faces and (2) that the sum of the face angles of any convex polyhedral angle must be less than 360°. (See $ 273.) Suppose first that each face is to be a triangle. Then, from the definition of a regular polyhedron, the triangle inust be equilateral. Each of its angles will therefore be 60° Consequently, by statement (2) above, polyhedral angles may be formed by combining three, four, or five such angles, but no more than five can be thus used, since six such angles amount to 360°, while seven or more of them exceed 360°. Therefore, not more than three regular polyhedrons are possible having triangles as faces. The three that are possible are the regular tetrahedron, regular octahedron, and regular icosahedron. (See Fig. 226.) Suppose secondly that each face is to be a square. Each face angle must then be 90° and, the sum of four such angles being 360°, it follows that but one regular polyhedron is possible having squares as sides. The cube is the one that is possible. Thirdly, suppose Ithat each face is to be a regular pentagon. Since each of the angles of such a figure is 108°, it follows that no more than one regular polyhedron is possible whose faces are pentagons. The one that is possible is the dodecahedron. We can proceed no farther, for the sum of three angles of a regular hexagon is 360°, while the sum of three angles of any regular polygon of more than six sides is greater than 360°. Hence the theorem is proved. NOTE. These regular solids occur in nature in the forms of a variety of crystals; but not all crystals are regular solids.

·278 POLYHEDRONS (VII, § 334 334. Models. Models of the five possible regular polyhedrons can be easily constructed as follows: Draw diagrams on cardboard as indicated in the figures below. Cut these out and then cut half way through the dotted lines so as to make it easy to fold along these lines. #1 A FIG. 227 Fold on the dotted lines so as to bring the edges together, subsequently pasting strips of paper over the edges to hold the solid in position. Models of the tetrahedron, octahedron, and icosahedron may also be made very quickly by hinging together short umbrella wires by means of strong copper wires strung through the end holes, joining together at each corner the proper number of rods. The student may show that each of these models will be quite rigid when completed.

·VII, § 335) GENERAL THEOREMS 279 335. Theorem XXI. Cavalieri's Theorem. If two solids are included between the same pair of parallel planes, and if every section of one of the solids by any plane parallel to one of these parallel planes is equal in area to the section of the other solid by the same plane, the volumes of the two solids are equal. Fig. 228 Outline of Proof. The two solids may be divided into a large number of thin slices by sections parallel to the two including planes. These slices may be thought of as approximately cylindrical, and the sum of all the slices in either case is the volume of the solid. The bases of two corresponding slices of the two solids between the same two planes are equal in area, by hypothesis, It is therefore apparent that the volumes of the two corresponding slices differ as little as we please if their thickness is sufficiently small. It can be shown in a precise manner that the total volume V of one of the solids, which is the sum of all such slices, differs from the total volume V' of the other solid by as little as we please. Hence, as in $$ 303, 311, it follows that V=V. Note. Observe also that $8 293, 303 are essentially special cases of what precedes.

·280 POLYHEDRONS [VII, § 336 336. Theorem XXII. The Prismoid Formula. If any solid S of any of the kinds considered in this Chapter is bounded by two parallel plane sections B and T, and if M denotes the area of another section parallel to and milway between B and T, the volume V of S is given by the formula: V= (B + T +-4 M).h/6, where B, T, and M denote the areas of the sections, and h denotes the distance between B and T. The proof of the preceding formula consists in showing that it reduces in every case to the very formula for volume that has already been proved in the articles above. Outline of Proof for Prisms and Cylinders. In these cases, all parallel sections are equal ($$ 276, 310). Hence B=T= M; and the formula to be proved becomes V=Bih, which we have already proved to be correct (S$ 296, 314). Outline of Proof for Pyramids and Cones. In these cases we know that the area of any section parallel to the base B is proportional to the square of its distance from the vertex ($$ 301, 322). Hence, since M is at a distance h/2 from the vertex, M_(h/2)2_1 or B=4 M. В. h2 4 The top section T is zero, since the top bounding plane meets a pyramid or a cone in just one point on the vertex. Hence the formula to be proved becomes, in this case, V=[B+T+4M]. h/6=(B+0+B). h/6= Bh/3, which we know to be correct. $$ 306, 323 Outline of Proof for Frustums. Given a frustum of a pyramid or of a cone, let H be the distance from the vertex O to the larger of the two bounding sections. Let B represent this larger section. Then H-h is the distance from 0 to the other bounding section T, and H - h/2 is the distance from 0 to the middle section M. We know that the volume V of the frustum is V=[BH - T(H-h)]/3. 88 307, 324 Or, since T/B=(H-h)/H, $8 301, 322 we know that si w ·VII, § 337] GENERAL THEOREMS 281 B V = B[11 -(II - h)/11?]/3 [3 H2-3 Hh + h]h/3. H2 Since T/B =(H - n°/H, and M/B =(II - h/2)/H4, the formula to be proved may be written, (H - h)? V =[B+ T + 4 M]h/6 = B+ h (H - h/2)2 B+4 B 112 II? B h [3 H2 - 3 Hh + 12] H2 This is equivalent to the formula that we know to be correct; hence the theorem is proved. 337. Uses of the Prismoid Formula. The prismoid formula is a convenient means of remembering the volumes of a variety of solids. We shall see in Chapter VIII that it holds for spheres and frustums of spheres as well as for the solids of this chapter. It also holds for any solid bounded by two parallel planes, made up by joining together pyramids, prisms, etc., bounded by the same two planes ; such a solid is called a prismoid. FIG. 229 The forniula is used very extensively by engineers to estimate the volumes of various objects, such as the volume of a hill, or the volume of a metal casting. Since the same formula holds for such a large variety of solids, it is reasonably safe to use it without even stopping to see which of these solids really resembles the object whose volume is desired. It is shown in more advanced books that the same formula holds whenever the area of the section by any plane parallel to the bounding planes is proportional to the square of the distance from some fixed point to that plane. Many solids not mentioned otherwise in elementary geometry satisfy this requirement.

·282 POLYHEDRONS [VII, § 337 MISCELLANEOUS EXERCISES. CHAPTER VII 1. Show that every section of a cylinder made by a plane passing through an element is a parallelogram. What is the section when the cylinder is a right cylinder ? 2. Show that every section of a cone made by a plane through the vertex is a triangle. What is the section when the cone is a right cone? 3. If a, b, and c are the dimensions of a parallelepiped, show that the length of its diagonal is Va? +62 +0. 4. How long an umbrella will go into' a trunk measuring 32 in. by 17 in. by 21 in., inside measure, (a) if the umbrella is laid on the bottom ? (b) if it is placed diagonally between opposite corners of the top and bottom ? 5. Find the volume of a pyramid whose base is a rhombus 6 in. on a side and whose height is 6 in., if one angle of the rhombus is 60°. 6. The Great Pyramid in Egypt is about 480 ft. high and its base is a square measuring about 764 ft. on a side. Find approximately its volume in cubic yards. 7. Water is poured into a cylindrical reservoir 25 ft. in diameter at the rate of 300 gallons a minute. Find the rate (number of inches per minute) at which the water rises in the reservoir (1 gal. = 231 cu. in.). 8. A copper teapot is 97 in. in diameter at the bottom, 8 in. at the top, and 11 in. deep. Allowing 42 sq. in. for locks and waste, how much metal is required for its construction, excluding the cover ? 9. A conical spire has a slant height of 60 ft, and the perimeter of the base is 50 ft. Find the lateral surface. 10. How many cubic inches of lead are there in a piece of lead pipe 2 yd. long, the outer diameter being 2 in, and the thickness of the lead being * of an inch ?

·-- VII, $ 337] MISCELLANEOUS EXERCISES 283 11. The chimney of a factory has approximately the shape of a frustum of a regular pyramid. Its height is 75 ft. and its upper and lower bases are squares whose sides are 5 ft. and 8 ft. respectively. The flue is throughout a square whose side is 3 ft. How many cubic feet of material does the chimney contain ? Assuming that a brick is 8 in. long, 3 in. wide, and 24 in. thick (as is ordinarily the case), estimate the number of bricks in such a chimney. 12. Compare the lateral areas, the total areas, and the volunes of (1) a right circular cylinder and a right circular cone having equal bases and altitudes, (2) a regular pyramid and a regular prism having congruent bases and equal altitudes. 13. A standard rain-gauge is made by inclosing a tube B in the interior of a can ACDE and connecting the mouth of the tube to the mouth of the can by a funnel FGHI. The amount of water, measured in inches (depth), that has fallen in the vicinity of the gauge is determined by reading the height of the water in the tube B. Find a formula for the amount of rain that has fallen in terms of the height h of the water in the tube B, the radius 1 of the tube, and the radius R of the can. Ans. hr2/R? 14. If one of the edges of a tetrahedron is 1 in. long, how long will be the corresponding edge of a similar tetrahedron of 8 times the volume? Answer the same question for the case in which the new tetrahedron is to have half the volume of the original Ans. 2 in.; 1/2 =.79 in. 15. It is usual to state the diameter d of a tube in inches, and the area A of its surface in square feet. Show that the formula used by engineers : 0.2618 di gives very nearly the correct value in square feet, if d is measured in inches, and the length 1 is measured in feet. E .

·CHAPTER VIII THE SPHERE PART I. GENERAL PROPERTIES 338. Spheres. A sphere is a portion of space bounded by a surface such that all straight lines to it from a fixed point within are equal. The fixed point within the sphere is called its center; line segment joining the center to any point on the surface is a OZ P' Fig. 230 . a radius; a line segment drawn through the center and terminated at both ends by the surface is a diameter. It follows from these definitions that: (a) The radii of a sphere, or equal spheres, are equal. (6) The diameters of a sphere, or of equal spheres, are equal. ©) Spheres having equal radii, or equal diameters, are equal. (d) A sphere may be generated by the revolution of a semicircle about its diameter. 284 ·VIII, $ 339) GENERAL PROPERTIES 285 EXERCISES Ax. 10.) 1. What is the locus of the points that are 2 in. from the surface of a sphere whose radius is 4 in. ? 2. Show that the distance from the center of a sphere to a point ontside the sphere is greater than the radius. (Use State the converse. Is it true? 3. If two sphores have the same center, they are called concentric. Show that one of two concentric spheres lies wholly within the other. 4. Show that if the center of each one of two given spheres lies on the surface of the other, their radii are equal. 5. Show by $ 77, that a plane perpendicular to a diameter of a sphere at its extremity has only one point in cominon with the sphere. o М. N 339. Tangent Planes and Lines. A plane that has only one point in common with a sphere is called a tangent plane to the sphere. A line that has only one point in common with a sphere is called a tangent line to the sphere. In either case, the single common point is called the point of tangency.

·286 THE SPHERE (VIII, § 340 340. Theorem I. A plane perpendicular to a diameter of a sphere at one of its extremities is tangent to the sphere. Outline of Proof. Let MN be a plane perpendicular to a diameter PP at P. Connect any point A of MN to the center of the sphere O. Show, by $ 77, that OA> OP; whence, by $338, A cannot be on the sphere, so that P is the only point the plane on the sphere. 341. Corollary 1. (Converse of Theorem I.) If a plane is tangent to a sphere, it is perpendicular to the radius drawn to the FIG. 231 point of contact. [Hint. Show, by Ax. 10, that the radius is shorter than any other line drawn from the center to the plane. Then use $77.] 342. Corollary 2. A straight line perpendicular to u diameter of a sphere at one of its extremities is tangent to the sphere; and conversely. [Hint. Use $$ 254, 340 for direct, and § 110 for converse.] 343. Corollary 3. All of the straight lines tangent to a sphere at a given point lie in the plane tangent to the sphere at that point. [Hint. Use $ 254.] N EXERCISES 1. What is the locus of a point in space at a given distance from a given point? 2. Prove that if two lines are tangent to a sphere at the same point, their plane is tangent to the sphere. [Hint. Connect this with one of the corollaries on this page.] 3. All lines tangent to a sphere from the same point are equal. (Hint. Connect the center of the sphere with the given point aud with two or more points of tangency.]

·VIII, § 344) GENERAL PROPERTIES 287 344. Theorem II. Every section of a sphere made by a plane is a circle. Fig. 232 Given the sphere whose center is 0, cut by a plane in the section AMBN. To prove that section AMBN is a circle. Proof. Draw OQI section AMBN; join Q to C and D, any two points in the perimeter of the section; draw OC and OD. In the rt. A OQC and OQD, 0Q=0Q, and OC= 0D. Why? Therefore A OQCA OQD. Why? Therefore QC = QD. Why? Since C and D are any two points on the perimeter of the section, all points on the perimeter of the section are equally distant from Q. Therefore section AMBC is a circle. § 103 EXERCISES 1. If, in Fig. 232, the radius of the sphere, OC, is 10 in., and the distance OQ from the center 0 to the plane AB is 6 in., find the radius CQ of the circle. 2. If, in Fig. 232, the distances CQ and OC are given, show how to find the distance OQ.

·288 THE SPHERE (VIII, $ 345 345. Great and Small Circles. A circle on the sphere whose plane passes through the center is called a great circle of the sphere, as CD, Fig. 233. FIG. 233 A circle on the sphere whose plane does not pass through the center is called a small circle of the sphere, as AB, Fig. 233. The axis of a circle of a sphere is the diameter of the sphere which is perpendicular to the plane of the circle. The poles of a circle of a sphere are the extremities of the axis of the circle, 346. Corollary 1. Through any three points on the surface of a sphere one and only one circle of the sphere may be drawn. [Hint. Use 4, § 241.] 347. Corollary 2. Through any two points on the surface of a sphere a great circle may be drawn. It follows from 4, § 241, that there is one and only one such great circle through the two given points, unless they lie at the opposite ends of a diameter. 348. Distance on a Sphere. By the distance between two points on the surface of a sphere is meant the length of the shorter arc of the great circle joining them. It can be shown that this is the shortest path on the surface of the sphere between the two points.

·VIII, § 348] GENERAL PROPERTIES 289 EXERCISES w 7. 1. If we consider the earth as a NE sphere, what kind of circles are the parallels of latitude ? the equator ? the meridians ? B 2. Prove that the axis of a small Ecircle of a sphere passes through the center of the circle; and conversely, a diameter of the sphere through the center of a small circle is the axis of that small circle. 3. Prove that in the same sphere, or in equal spheres, all great circles are equal. 4. The radius of a sphere is 10 in. Find the area of a section made by a plane 5 in. from the center. 5. The area of a section of a sphere 7 in. from the center is 288 7 sq. in. Find the area of a section 4 in. from the center. 6. Prove that in the same sphere, or in equal spheres, if two sections are equal, they are equally distant from the center, and conversely. 7. Prove that any two great circles of a sphere bisect each other.

·290 THE SPHERE (VIII, $ 349 349. Theorem III. All points in the circumference of a circle of a sphere are equally distant from either one of its poles. p' FIG. 234 (vr) Fig. 234 (6) Given any two points A and B in the circunference of the circle ABC, and P and P', the poles of ABC. To prove that PA = PB, and Pl1=P'B. Proof. Draw the great circles PAP' and PBP'. Let D be the intersection of the axis PP' with the plane ABC. Draw the straight lines AD, BD, PA and PB. Now PD= PD, and DA= DB, Why? and LPDA=% PDB= 90°. Why? Hence chord PA=chord PB, Why? Therefore PA=PB. Why? In the same way it may be proved that P'A=P'B. Note. The manner in which circles may be drawn on a sphere is illustrated by Fig. 234 (6). If one end of a string is held at any point on the sphere, while a pencil attached to the other end is moved around the spbere, keeping the string taut, the end of the pencil describes a circle on the sphere, by Theorem III. The various figures drawn in this chapter can be reproduced on the surface of an actual sphere, by this method of drawing the circles.

·VIII, § 351] GENERAL PROPERTIES 291 350. Polar Distance. The polar distance of a circle of a sphere is the distance on the sphere (§ 348) from its nearest pole to any point of the circumference, as PA or PB in Fig. 234. A quadrant is one fourth part of the circumference of a great circle; i.e. an arc of 90° on a great circle. 351. Corollary 1. quadrant. The polar distance of a great circle is it P Fig. 230 [Hint. Let ABC be a great circle. Then its center is also the center of the great circle PBP'. Hence the arc PB measures the right angle POB.) EXERCISES 1. What is the locus of all the points on the surface of the earth at a quadrant's distance froin the north pole ? from the south pole ? from the equator ? See the figure for Ex. 1, p. 289. 2. The distance of the plane of a certain small circle from the center of a sphere is one half the radius of the sphere. If the diameter of the sphere is 12 in., find the polar distance of the small circle in degrees and in inches. Ans. 60°; 2* in. 3. Show that a great circle on the earth whose poles lie on the equator passes through the north pole.

·292
THE SPHERE
[VIII, § 352
352. Problem I.
sphere.
To determine the radius of a given material
D'a
A of
QP
FIG. 236 (u)
FIG, 236 ()
Given any material sphere, OPQ.
To find its radius.
Construction. Take any point P on the surface of the sphere
as a pole, and describe the circle ABC.
Take any three points on this circle, as A, B, C.
By means of the compasses construct on paper or on the
black board the triangle A'B'C' congruent to the triangle
ABC.
Circumscribe a circle around A A'B'C', and let D' be the
center of this circle.
Draw D"A" equal to the radius D'A'.
Through D" draw an indefinite line P'Q' perpendicular, to
D'A".
From A" lay off with compasses A”P' equal to line AP.
At A" erect a perpendicular to 4"P' and extend it to meet
P'Q at Q.
Then P'Q is the diameter and P'Q/2 is the radius of the
given sphere.
[The proof is left to the student. 7

·VIII, 8 353]
GENERAL PROPERTIES
293
353. Problem II. To construct a sphere through four given
points not all in the same plane.
11
Fig. 237
Given the four points A, B, C, D not all in the same plane.
To construct a sphere that passes through A, B, C, and D.
Construction. At E, the middle point of AB, erect a plane
QEP perpendicular to AB. Likewise, let PFR be a plane
perpendicular to BC at its iniddle point F; and let QGR be a
plane perpendicular to BD at its middle point G.
Let O be the point common to all three planes QEP, PFR,
and QGR.
With O as center, and 04 as radius, draw a sphere.
This is the required sphere passing through A, B, C, and D.
Proof. The plane QEP is the locus of all points equidistant
froin A and B.
Why?
Likewise, PFR is the locus of points equidistant from B
and C; and QGR is the locus of points equidistant from B
and D.
The planes QEP and PFR meet in a line OP. Why?
The line OP meets the plane QGR in a single point 0.
Why?
Therefore is equidistant from A, B, C, D.
Why?
Moreover, O is the only point equidistant from A, B, C, D.
Why?

·294 THE SPHERE (VIII, $ 354 354. Inscribed and Circumscribed Spheres. A sphere is said to be circumscribed about any polyhedron when the vertices of the polyhedron all lie on its surface. A FIG. 238. CIRCUMSCRIBED SPHERE A sphere is said to be inscribed in any polyhedron when it is tangent to each of the faces of the polyhedron. B. FIG. 239. INSCRIBED SPHERE 355. Corollary 1. One and only one sphere may be circumscribed about any given tetrahedron (triangular pyramid). [Hint. Pass a sphere through the four vertices as in 353. Notice that the four vertices cannot all lie in one plane.]

·VIII, § 356] GENERAL PROPERTIES 295 356. Problem III. To inscribe a sphere in a given tetrahe dron (triangular pyramid). Given the tetrahedron ABCD, Fig. 239. To construct the sphere inscribed in it. Construction and Proof. Bisect the dihedral angles whose edges are BC, CD, and DB, by the planes BOC, CED, and DEB, respectively. The plane BOC is the locus of the points equidistant from the faces BCD and BAC'; the plane CED is the locus of the points equidistant from the faces BCD) and CAD; and the plane DEB is the locus of the points equidistant from the faces BCD and DAB. Why? The intersection of these planes is equidistant from the four faces of the tetrahedron. Hence the sphere whose center is O and whose radius is the perpendicular distance OE from O to the face ABC, is tangent to each of the faces; it is therefore inscribed in the tetrahedron. No other sphere exists that is inscribed in the tetrahedron, for no other point than 0 is equiclistant from the four faces. Why? EXERCISES 1. By means of an instrunent called a spherometer, the distances AD and DP, Fig. 236, can be measured directly. Show, by $ 162, how to find the radius from these values. 2. Show that the process of $ 352 can be used to find the radius of a sphere, if only a piece of the sphere is available, as in the case of a glass lens. 3. Show that four points in space determine a sphere, provided they do not lie in one plane. 4. Show that a sphere is determined if any circle that lies on it and one pole of that circle are given. 5. Show that any two circles of a sphere completely determine the sphere.

·296 THE SPHERE (VIII, $ 357 PART II. SPHERICAL ANGLES TRIANGLES POLYGONS 357. Spherical Angles. The line tangent to a great circle of a sphere at any point is a tangent to the sphere at that point; for it touches the sphere in only one point ($ 339). The angle formed by the intersection of two great circles is called a spherical angle. It is defined to be equal to the angle formed by the tangents to the two great circles, at their point of intersection, as the angle CPD, Fig. 240. 358. Theorem IV. The angle between two great circles is measured by the arc of a great circle described from its vertex as a pole and included between its sides. M PY B P Fig. 240 Given the great circles PAP and PBP" intersecting at P, and AB the arc of a great circle described with P as a pole. To prove that AB is the measure of Z APB. Proof. Draw the radii 04, OB, and the tangents PC, PD. Then OA || PC and OB | PD. Why? Hence ZAOB LCPD. Why? But 2 AOB is measured by the arc AB; hence 2 CPD is ·VIII, 360) TRIANGLES AND POLYGONS 297 ineasured by the arc AB. It follows that the spherical angle APB, which is equal to L CPD by definition ($ 357), is measured by the arc AB.. 359. Corollary 1. The spherical angle between two great circles is equal to the plane angle of the dihedral angle formed by the planes of the two great circles. 360. Spherical Triangles and Polygons. A spherical polygon is à portion of a spherical surface bounded by three or more arcs of great circles; as ABCDE, Fig. 241. The bounding ares of great circles are called the sides of the spherical polygon; their intersections, the vertices; and the angles formed by the sides at the vertices, the angles of the spherical FIG. 241 polygon. A diagonal of a spherical polygon is an arc of a great circle joining any two non-adjacent vertices. A spherical triangle lis a spherical polygon of three sides, as ABC, Fig. 242. The words isosceles, equilateral, acute, right, and obtuse are applied to spherical triangles in precisely the same way as to plane triangles. Thus, in Fig. 242, the spherical triangle ABC is isosceles if the two sides, as AB and BC, are equal; the triangle is equilateral if AB=BC = AC; the triangle is a right triangle if any one angle is a right angle; etc. FIG. 242 ·298 THE SPHERE [VIII, $ 361 361. Relation to Central Polyhedral Angles. The planes of the arcs of the great circles forming the sides of a spherical polygon meet at the center of the sphere and form a polyhedral angle, as 0-ABCD. Fig. 243 This polyhedral angle and the spherical polygon are so closely related that the student can easily prove the following statements : (a) The sides of a spherical polygon have the sume meusures as the corresponding face angles of the polyhedral angle. (6) The angles of the spherical polygon have the same measures as the corresponding dihedral angles of the polyhedral angle. Thus, sides AB, BC, etc., of the spherical polygon ABCD have the same measures as face 4 AOB, BOC, etc., of polyhedral 20-ABCD; and spherical & ABC, BCD, etc., have the same measures as the dihedral As whose edges are OB, OC, etc. (c) Any angle of a spherical polygon (or, the corresponding dihedral angle of the polyhedral angle) is measured by the arc of a great circle described with the vertex of the angle as pole and terminated by the sides. See $$ 358, 359. In general, any fact proved for the sides and the angles of a spherical polygon is true also for the corresponding face angles and dihedral angles of the corresponding central polyhedral angle.

·VIII, § 363] TRIANGLES AND POLYGONS 299 362. Theorem V. The sum of any two sides of a spherical triangle is greater than the third side. [Compare $ 272.] Fig. 244 $ 272 Given the spherical A ABC. To prove that AB + BC > CA. Proof. LA0B + BOC>ZCOA LAOB is measured by ÁR, 2 BOC is measured by BO, ZCOA is measured by A. Therefore AB #BOXCA. Why ? 363. Theorem VI. The sum of the sides of any convex spherical polygon is less than 360°. [Compare $ 273.] Given the spherical polygon ABCD. To prove that AB + BC + CD + [HINT. Make use of $ 273.) Da < 360°. FIG. 246 ·300 THE SPHERE (VIII, $ 364 EXERCISES 1. Show that any side of a spherical polygon is less than 180°. 2. In the spherical A ABC, AB = 35', and BC = 75°. Between what limits must A lie? 3. Three of the sides of a spherical quadrilateral are respectively 88° 17', 70° 36', and 50° 33'. Between what limits must the fourth side lie ? 364. Polar Triangles. If from the vertices of a spherical triangle as poles arcs of great circles are drawn, these arcs forin a second triangle which is called the polar triangle of the first. a FIG. 246 Thus, if A, B, C, the vertices of the spherical A ABC, are the poles of the arcs B'C", A'C", A'B', forming the spherical A A'B'C', then A'B'C' is the polar triangle of ABC. If the entire circles be drawn, they will intersect so as to form eight spherical triangles, but the polar of the given triangle ABC is that one of the eight triangles whose vertices lie on the same side of the arcs of the given triangle as the corresponding vertices of the given triangle, and no side of which is greater than 180°.

·VIII, § 365]
TRIANGLES AND POLYGONS
301
365. Theorem VII. If one spherical triangle is the polar of
another, then the second is the polar of the first.
FIG. 247
Given A A'B'C', the polar of A ABC.
To prove that ABC is the polar of A l'B'C'.
Proof. A is the pole of B'C", and C is the pole of A'B';
Given.
hence B' is at a quadrant’s distance from A and C, so that B' is
the pole of C.
$ 249
Similarly, 1' is the pole of BC, and C' is the pole of AB.
Therefore ABC is the polar triangle of A'B'C'.
$364.
EXERCISES
1. Show that if one side of a spherical triangle on the earth's
surface is on the equator, one vertex of the polar triangle is either
at the north pole or at the south pole.
2. Show that if one vertex of a triangle on the earth is at
the north pole, one side of the polar triangle is on the equator.
3. Show that if one vertex of a triangle on the earth is at
the north pole, and if one side of the triangle is on the equator,
the polar triangle also has one vertex at the north pole and
one side along the equator.

·302 THE SPHERE (VIII, § 366 366. Theorem VIII. In two polar triangles, each angle of the one is measured by the supplement of the side opposite to it in the other. FIG, 248 Given the polar triangles ABC and A'B'C', with the sides denoted by a, b, c, and a, b, c', respectively. To prove that (a) LA + a' = 180°, LB+W=180°, 20+d=180°; (6) LA' + a = 180°, ZB'+b=180°, 20'+c=180°. Proof. Let AB and AC (prolonged, if necessary) intersect BC at D and E, respectively. Then CD=90°, and ÉB' = 90°. Why? Therefore C") + BB'=180°. Why? That is CE + ED+ED + DB' = 180°, or, ÉD ta'=180°. But ED is the measure of 2 A. (C) $ 361 Therefore LA+a'=180°. In a similar manner ZB+0=180°, and 2 C +=180°. The proof of (b) is left for the student. EXERCISE 1. If the angles of a spherical triangle are 70°, 90°, and 80°, respectively, find the sides of the polar triangle (in degrees).

·VIII, § 368] TRIANGLES AND POLYGONS 303 367. Theorem IX. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. FIG. 249 Given the spherical A ABC with the sides a, b, and c. To prove that ZA+ZB+2C> 180° and < 540°. Proof. Let A A'B'C', with its sides denoted by a', , and c', be the polar of A ABC. Then LA + '=180°, LB+W = 180°, 20+d=180°. Therefore LA+ZB+20+d+W+d=540'. Why? But a + B +0 < 360°. § 363 Therefore Lo+ZB+20> 180°. Why? Again (t' + B + > 0°. Therefore LA+LB+ LC < 540°. Why? 368. Corollary 1. In a spherical triangle there can be one, two, or even three right angles; there can be one, two, or three obtuse angles. EXERCISES 1. Show that a triangle on the earth's surface whose sides are the equator and two me lians, has two of its angles right angles, and two of its sides quadrants. 2. If, as in Ex. 1, two of the angles of a spherical triangle are right angles, between what limits must the third angle lie?

·304 THE SPHERE (VIII, $ 369 369. Birectangular and Trirectangular Triangles. A spherical triangle having two right angles is called a birectangu- FIG. 250 lar spherical triangle. A spherical triangle having all of its angles right angles is called a trirectangular spherical triangle. If ZP in the figure is either acute or obtuse, while 4A and B are right, A ABP is birectangular; if ZP is also a right angle, A ABP is trirectangular, as in Fig. 230, p. 284. EXERCISES 1. The sides of a spherical triangle are 80°, and 126°, and 175°. How large are the angles of its polar triangle? 2. Show that in a birectangular triangle the sides opposite the right angles are quadrants. 3. Show that three mutually perpendicular planes through the center of a sphere divide its surface into eight congruent trirectangular triangles. 4. Show that the area of a trirectangular triangle on a sphere is one eighth of the area of the sphere. 5. Show that each of the sides of a trirectangular triangle is a quadrant. Hence show that the polar of a trirectangular triangle coincides with it.

·VIII, S 371) TRIANGLES AND POLYGONS 305 370. Symmetric Triangles. Two spherical triangles are symmetric when their parts are equal each to each, but are in opposite order. Thus, in the S ABC and A'B'C' (Fig. 251), B FIG. 251 if angles A= A', B=B', C -- C", and sides AB= A'B', BC= B'C', CA=C'A', but the order of arrangement is opposite in the two figures, the triangles are symmetric. In general, two symmetric triangles cannot be superposed and hence cannot be said to be congruent. Thus, if A ABC is moved so that side AB coincides with its equal, A'B', in the syinmetric A A'B'C', then the vertices C and Clie on opposite sides of A'B'. In plane triangles, A ABC could be revolved about AB till it coincided with A A'B'C'; but this is in general impossible with spherical triangles. 371. Corollary. Two isosceles symmetric spherical triangles are congruent. EXERCISES 1. Prove that the base angles of an isosceles spherical triangle are equal. [IIInt. Draw an arc bisecting the vertical angle, thus forming two symmetric triangles.] 2. Show that if two sides of a spherical triangle are quadrants, the triangle is birectangular.

·306 THE SPHERE (VIII, & 372 372. Theorem X. Two triangles on the same sphere, or on equal spheres, are either congruent or symmetric, if two sides and the included angle of the one are equal, respectively, to two sides and the included angle of the other. FIG. 252 (a) Fig. 252 (0) Given the spherical A ABC and A'B'C' on the same sphere or equal spheres, having AB= A'B', AC= A'C", LA=L A'. To prove that A ABC and A'B'C" are either congruent or else symmetric. Proof. If the equal parts of the two triangles are in the same order, A ABC can be placed on A A'B'C" as in the corresponding case of plane triangles. See Fig. 252 (a). If the equal parts of the two triangles are not in the same order, construct A A'B'C" symmetric to A A'B'C'. (Fig. 252 (b).) In A ABC and A'B'C", AC= AC", AB = A'B', and Z 1 2 B'AC". Since these parts are arranged in the same order, A ABC and A'B'C" are congruent. Therefore spherical A ABC is symmetric to spherical A A'B'C'. Why? 373. Theorem XI. Two triangles on the same sphere, or on equal spheres, are either congruent or symmetric, if two angles and the included side of the one are equal, respectively, to two angles and the included side of the other. [Proceed as in $ 372.]

·VIII, § 375] TRIANGLES AND POLYGONS 307 374. Theorem XII. Tuo triangles on the same sphere, or on erual spheres, are either congruent or symmetric, if the three sides of the one are equal, respectively, to the three sides of the other. [The proof is left to the student. ] 375. Theorem XIII. Two triangles on the same sphere, or on equal spheres, are either congruent or symmetric, if the three angles of the one are equal, respectively, to the three angles of the other. FIG. 253 Outline of Proof. If ABC and A'B'C" are the two given spherical triangles so that LA=LA, ZB=2 B', 2C=2C", their polar triangles LMN and L'M'N' have the three sides of one equal to the three sides of the other, respectively. § 366 Then, by $ 374, A LMN and I'M'N' are either congruent or symmetric. In either case, the three angles of ALMN are equal to the three angles of A L'M'N', respectively; and therefore the three sides of A ABC are equal to the three sides of A A'B'C', respectively. § 366 It follows, by $ 374, that S ABC and A'B'C' are either congruent or symmetric. Note. Theorems analogous to those of $$ 41, 43, 44, etc., may be proved in a manner similar to $$ 372-375.

·308 THE SPHERE (VIII, 8 375 EXERCISES 1. Show that two trihedral angles are congruent if they intercept congruent triangles on the surfaces of two equal spheres whose centers are at their vertices, respectively. 2. If two trihedral angles intercept symmetric spherical tri. angles on the surface of a sphere whose center is at their vertices, respectively, show that the face angles and the dihedral angles of one trihedral angle are equal to those of the other, but taken in reversed order. [Such trihedral angles are called symmetric.] 3. Prove the following theorein, which states for trihedral angles ($ 361) a theorem analogous to that of $ 372: Two trihedral angles are either congruent or symmetric if two face angles and the included dihedral angle of the one are respectively equal to two face angles and the included dihedral angle of the other. (Hint: Consider the spherical triangles cut out by the two trihedral angles on the surfaces of two equal spheres whose centers lie at the vertices of the two trihedral angles, and apply $ 372.] 4. Prove the following theorem, analogous to $ 373: Two trihedral angles are either congruent or symmetric if two dihedral angles and the included face angle of the one are respectively equal to two dihedral angles and the included face angle of the other. .. .. 5. State and prove theorems for trihedral angles similar to Theorems XII-XIII, § 374–375. 6. Prove Theorem XI, § 373, by first considering, as in $ 375, the polars of the given triangles, and applying $ 372. 7. Show that any trirectangular triangle on the earth's surface is congruent to the trirectangular triangle formed by the equator and two meridians whose longitude differs by 90°.

·VIII, § 376) AREAS AND VOLUMES 309 PART III. AREAS AND VOLUMES 376. Theorem XIV. The area of the surface generated by a straight line revolving about an axis in its plane is equal to the product of the projection of the line on the axis and the length of the circle whose radius is a perpendicular erected at the middle point of the line and terminated by the axis. B А с X Y E F DO FIG. 254 Given EF, the projection upon XY of AB revolving about XY, and OP I AB at its mid-point, and meeting XY at 0. To prove that the area generated by AB EF x 2 пOP. Proof. Draw PD I XY, and AC|XY. Since the surface generated by AB is the lateral surface of the frustum of a cone, the area generated by AB is AB ME + BF 2 TAE #2 7 BF ABX 25 2 2 = AB X 2 PD. $ 321 Now A ABC ~ A POD. $ 157 Therefore AB: OP= AC:PD. Why? Then AB X PD: AC X OP= EFX OP. Why? And AB X 2 *PD = EF X20P. That is, the area generated by AB is EF X 2 OP. If AB meets XY, the surface generated is a conical surface whose area again EF X20P. $ 320 If AB is parallel to XY, the surface generated is a cylindrical surface whose area again EF X2 +OP. § 313 ·310 THE SPHERE [VIII, § 377 377. Theorem XV. The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle. AF GE Fig. 255 Given a sphere generated by the revolution of the semicircle . ABCDE about the diameter A0E, S being the area of the surface, r being the radius, and d being the diameter. To prove that S 2 mrd. Proof. Inscribe in the semicircle half of a regular polygon ABCDE, of any number of sides, and draw BF, CO, DG, perpendicular to AE. From O draw OPL AB. Then OP bisects AB, Why? and is equal to each of the drawn from 0 to the equal chords BC, CD, DE. Why? Now the area generated by AB= AF X2T.OP, $ 376 the area generated by BC FOX 2 • OP the area generated by CD= OG X 2 OP, the area generated by DE=GEX2 • ОР. Therefore, if S" denotes the surface generated by the semipolygon, S'=(AF+FO + OG + GE)2 OP= AEX 2 TOP. Let the number of sides of the semipolygon be now indefinitely increased. Then OP has for its limit r, the semipolygon for its limit the semicircle, and S' for its limit S. Hence, as in § 303, S = AE x 2 TT. 378. Corollary 1. The area of the surface of a sphere is equal to 4 mm ·VIII, § 382] AREAS AND VOLUMES 311 379. Corollary 2. The area of the surface of a sphere is equal to the sum of the areas of four great circles. For $= 2 r X 2 =472, $ 378 and tra is the area of a great circle. 380. Corollary 3. The areas of the surfaces of two spheres are to each other as the squares of their radii; or, as the squares of their diameters. 381. Zones. A zone is a portion of the surface of a sphere bounded by the circumferences of two circles whose planes are parallel. N А B" virpinginan FIG. 256. ZONES ON THE EARTH'S SURFACE The circumferences forming the boundary of a zone are its bases. If the semicircle NES is revolved about NS as an axis, arc AB will generate a zone, while points A and B will generate the bases of the zone. The altitude of a zone is the perpendicular distance between the planes of the bases. 382. Corollary 4. The area of a zone of a sphere is equal to the product of the altitude h of the zone and the circumference of a great circle; or 2 trh, where r is the radius of the sphere.

·312
THE SPHERE
(VIII, § 383
383. Lunes. A. lune is a portion of a spherical surface
bounded by two semicircumferences of great circles; as
C
Fig. 257
ABCDA (Fig. 257). The angle of a lune is the angle formed
by its bounding arcs. Thus B.ID is the angle of the lune
ABCD4.
384. Theorem XVI. The area of a lune is to the area of the
surfice of the sphere as the angle of the lune is to four right
angles.
P
C
Fit, 258
Given the lune PAP'B, let L denote the area of the lune, S
the area of the surface of the sphere, and a the angle of the
lune.

·VIII, § 386] AREAS AND VOLUMES 313 To prove that L/S= 2/4 rt. A. Proof. With P as a pole describe the great circle ABCD. Then the arc AB measures La of the lune. Why? Therefore arc AB/circle ABCD=La/4 rt. A. If AB and ABCD are commensurable, let their common measure be contained m times in AB and n times in ABCD. Then are AB/circle ABCD=m/n. Therefore a./4 rt. 6 =m/n. $ 358 Pass arcs of great circles through each point of division of ABCD and the poles P and P'. These arcs will divide the entire surface into n equal lunes, of which PAPB will contain m. Therefore L/S= m/n, or, L/S= a/4rt. &. If AB and ABCD are incommensurable, the theorem can be proved as in § 130. The details are left to the student. 385. Corollary 1. The area of a lune whose (ingle is 1° is 4 772/360 = 72/90. 386. Corollary 2. The area of 4 lune whose angle is kº is 4 #7K/360= ark/90. EXERCISES 1. If the surface of a sphere is 10 sq. ft., what is the area of a lune whose angle is 40° ? What is the radius of the sphere? Ans. 1} sq. ft.; 0.89+ ft. 2. Show that two lunes on the same sphere or equal spheres have the same ratio as their angles. 3. What is the angle of a lune which has the same area as a trirectangular triangle? 4. Show that the area of a lune is one ninetieth of the area of a great circle multiplied by the number of degrees in the angle of the lune.

·314 THE SPHERE (VIII, § 387 387. Theorem XVII. Two symmetric triangles are equal in area. Given the two symmetric spherical triangles ABC and A'B'C'. To prove that A ABC A A'B'C". Proof. Let P be the pole of the small circle passing through the points 1, B, C, and draw the great circle ares PA, PB, and PC. Then PA= PB= PC. FIG. 259 Why? Now place the two triangles diametrically opposite to each other and draw the diameter POPI. Also draw the great circle arcs P'A', P'B', and PC". Then the triangles PBC and P'B'C" are symmetrical and isosceles and therefore congruent. $ 371. Similarly APCA ZA PC'A', B and A PABAPA'B'. That is, the three parts of ABC are respectively congruent to the three parts of 1BC. Therefore A ABC= A A'B'C'. 388. Corollary 1. If two semicircumferences of great circles BCB' FIG. 260 and ACA' intersect on the surface of a hemisphere, the sum of the areas of the two opposite spherical triangles ACB and A'CB' is equal to the area of a lune whose angle is equal to ACB. [Hint. Show that the triangle ABC is symmetric to the triangle A'B'C'. Hence show that A ACB + A ACB' = lune A' CB'C'.]

·VIII, § 391] AREAS AND VOLUMES 315 389. Spherical Degree. The area of a lune whose angle is 1° is 4 /360, or ar*/90 (8 385). Half this area, that is, 4 712/720 or Tre/180, is often taken as a unit of area on the sphere, and it is called a spherical degree. 390. Measure of Solid Angles. A trihedral angle whose vertex is at the center of a sphere cuts out a spherical triangle on the surface of the sphere. The area of the spherical triangle, in spherical degrees, is called the measure of the trihedral angle. Likewise any polyhedral angle is measured by the area, in spherical degrees, that it cuts out upon the surface of a sphere whose center is at its vertex. EXERCISES 1. Show that the area of a lune whose angle is 1° is 2 spherical degrees. Р 2. Show that the area of the entire sphere is 720 spherical degrees 3. Show that the area of a birectangular triangle whose third angle is 1° is 1 spherical degree. 4. Show that the area of a trirectangular triangle is 90 spherical degrees, or one eighth of the entire surface. AB 391. Spherical Excess. The excess of the sum of the angles of a spherical triangle over 180° is called the spherical excess of the triangle. If, for example, the angles of a spherical triangle are 80°, 100°, and 125°, the spherical excess of the triangle is 125º. Likewise, the spherical excess of any spherical polygon is the excess of the sum of its angles above the sum of the angles of a plane polygon of the same number of sides.

·316
THE SPHERE
[VIII, 8 392
392. Theorem XVIII. The area of a spherical triangle is
equal to the area of a lune whose angle is half the spherical excess
of the triangle.
Fig. 261
area
Given the spherical A ABC.
To prove that A ABC is equal to a lune whose angle is
(21+2B+2 0–1809).
Proof. Complete the great circles by producing the sides of
the A ABC, as in Fig. 261.
Since A 1B'C" and A'BC are syininetric, they are equal in
$ 387
Therefore lune ABC=A ABC+A1B'C". $ 388
But, denoting the area of the whole sphere by S,
A CBA+AAC'B + A ABC+A AB'C' = S. Why?
Therefore
(lune BCB'.1-A ABC)+(lune CACB-A ABC)
+lune ABA'C=S.
Why?
Therefore, transposing, we obtain
2 A ABC=lune ABAC+lune BCB'A + lune CACB -- 1S.
But S is the area of a lune whose angle is 180º.
Therefore A ABC is equal to a lune whose angle is
(24+2 B+ 2C-180°).
$ 384

·VIII, § 396] AREAS AND VOLUMES 317 393. Corollary 1. The area of a spherical triangle, meusured in sphericul degrees, is numerically equal to its spherical excess. NOTE. This result enables us to compute the area of any spherical triangle in ordinary units of area, when we know its angles and the radius of the sphere. Thus, if » denotes the radius of the sphere, E the spherical excess, and A the required area, we have, by $ 385 A = EX 712E 180 180 Пr22 394. Corollary 2. The area of a trirectangular triangle is 90 spherical degrecs. 395. Corollary 3. The area of any spherical triangle is to the area of the entire sere us its spherical edress is to 720º. 396. Solid Angles. In general, if any close<l polygon or curve is drawn on the surface of a sphere, the figure formed by all radii of the sphere that join the center to the points of this figure on the spherical surface is called a solid angle. The area on the surface of the sphere cut out by such a solid angle, in spherical degrees, is the measure of the solid angle. EXERCISES 1. What is the measure of a hemisphere in spherical degrees? 2. The radius of a spliere is 2 ft. Find the area of a triangle on its surface whose angles are 75°, 35°, 105°, respectively. Solve first by $ 392; then by $ 394. Ans. 77/9 sq. ft. 3. The radins of the earth is approximately 4000 miles. Find the entire areal. Show that the area in square miles of one spherical degree is approximately 278,000 square miles. 4. Find how large a triangle on the earth's surface would have the total sum of its three angles equal to 181º. 6. Show that a region containing about 270,000 sq. mi. on the earth contains no triangle whose spherical excess is 1º. 6. What is the area of the state in which you live? What is its ineasure in spherical degrees ?

·318 THE SPHERE (VIII, $ 397 397. Theorem XIX. The volume V of a sphere is equal to the product of its surface by one third of its radius ; or, V=4 7703/3. FIG. 202 Given a sphere whose center is 0); let S denote its surface, rits radins, and V its volume. To prove that V=Sxr/3 = 4 73/3 Proof. Circuinscribe about the sphere any polyhedron as D-ABC, and denote its surface by S' and its volume by V'. Forin pyramids, as (-ABC, etc., having the faces of the polyhedron as bases and the center of the sphere as a common vertex. These pyramids will have a common altitude equal to r, and the volume of each pyramid is equal to its base x r/3. Why? Therefore V' = S'X 7/3. Why? If the number of pyramids is indefinitely increased by passing planes tangent to the sphere at points where the edges of the pyramids cut the surface of the sphere, as in Fig. 262, the difference between S and S' becomes as small as we please; the difference between V and V' becomes as small as we please.

·VIII, $ 400] AREAS AND VOLUMES 319 But however great the number of Pyramids, V'=S' XT/3. Therefore, as in § 303, V=SXT/3. Since S= 4 +12, V=4 nr2 X 1/3 - 4 7 13/3. it follows that 398. Corollary 1. The volumes of two spheres are to each other as the cubes of their radii, or as the cubes of their diameters. 399. Corollary 2. The volume of the pyramidal piece cut out of a sphere by any polyhedral angle whose vertex is at the center is equal to one third the area of the spherical polygon cut out of the surface times the radius. 400. Corollary 3. The prismoid formula (336) holds for a sphere. Tilint. Two parallel planes that include the entire sphere are tangent planes at the ends of a diameter; these cut the sphere in only one point each. A plane parallel to these two and halfway between them cuts the sphere in a great circle. Hence, in the notation of $ 336, B=0, T= 0, M= 4 Tr?, h =2r; hence the prismoid formula would give 0 + 0 + 4 na 4 που 2 r V=n[B+ 71+ 4 M - [º+ which, by $ 397, is correct.] [Note. As a matter of fact, the prismoid formula holds for the portion of a sphere intercepted between any two parallel planes.] EXERCISES 1. Assuming that the earth is a sphere whose radius is 4000 mi., find its volume. 2. Show that a cube circumscribed about a sphere has a volume 8 gol. Hence show that the sphere occupies a little more than half this volume ·320 THE SPHERE (VIII, $ 400 3. Find the volume of the material in a hollow sphere, if the radius of the outer surface is 6 in, and that of the inner surface is 5 in. 4. Show that the volume of a hollow sphere whose outer and inner radii are R and r, respectively, is 4 (R - 7/3. 5. Find the volume of the material in a hollow sphere whose outer radius is 10 in., if the material is in. thick. 6. Show that the volume of a sphere in terms of its diameter, d, is 7d/6. 7. If the radius of one sphere is twice that of another, how do their volumes compare? 8. If the volume of one sphere is twice that of another, how do their radii compare? 9. Find approximately the radius of a sphere whose volume is 100 cu. in. 10. How many shot ito in. in diameter can be made from 10 cu. in. of lead ? 11. If oranges 3 in. in diameter sell for 30 cents per dozen, and those 4 in. in diameter sell for 50 cents per dozen, which are the cheaper by volume? 12. If the skins are of equal thickness, which of the oranges of Ex. 11 has the greater percentage of skin to the cubic inch of volume? 13. Assuming that raindrops are practically spherical, if the diameter of one drop is half that of another, how do their volumes compare their areas ? 14. Which of the two drops of Ex. 13 has the greater ratio of area to volume? How much greater? Which will fall the more rapidly through the air ? (Hint. The greater the ratio of area to volume, for the same material, the slower the body will fall through the air. ] ·VIII, § 400] AREAS AND VOLUMES 321 15. Explain, by the principle of Ex. 14, why very small dust particles remain floating in the air for a long time. 16. The same amount of material, in the form of a cube, is melted and cast into a sphere. Is the surface area less in the form of the cube or in that of the sphere? [Hint. Assume the cube to be 1 unit on each edge; find the radius of the resulting sphere.] 17. If a surveyor wishes to be certain that the sum of the angles in any triangle in a region on the earth's surface shall be equal to 180° to within one minute, how large may the region be? Ans. About 4600 sq. mi. 18. Demonstrate the existence of spherical triangles with three obtuse angles from the existence of triangles whose sides are very short.

·No text on this page.

·TABLES
TABLE I
RATION OF THE SIDES OF RIGHT TRIANGLES
and
CHORDS AND ARCS OF A UNIT CIRCLE
TABLE JI
SQUARES AND SQUARE ROOTS OF NUMBERS
CUBES ANI) CUBE ROOTS OF NUMBERS
TABLE III
VALUES OF IMPORTANT NUMBERS
including
UNITS OF MEASUREMENT

·TABLE I RATIOS OF THE SIDES OF RIGHT TRIANGLES AND LENGTHS OF CHORDS AND ARCS OF A UNIT CIRCLE EXPLANATION OF TABLE I 1. Ratios of the Sides of Right Triangles. If an angle given in the Angle Column is one acute angle of a right triangle : The Sine Column gives the ratio of the side opposite the angle to the hypotenuse; The Tangent Column gives the ratio of the side opposite the angle to the side adjacent to the angle. To find the Cosine of any angle, take the sine of the complement of that angle. 2. Chords and Arcs of a Unit Circle. If an angle given in the Angle Column is an angle at the center of a circle of unit radius: The Chord Column gives the length of the chord that subtends that angle; The Arc Column gives the length of the arc that subtends that angle. To find the lengths of chords or arcs of any circle of radius r, multiply the values given in the table by that radius. The table is limited to angles less than 90°; but to find the chord that subtends an obtuse angle, first take half the angle, find the sine of this half angle, and multiply by 2. This follows from the fact that the chord of any angle is twice the sine of half that angle.

·السا Quantities Determined by a Given Angle iii Angle Sine Tangent Chord Arc Angle Sine Tangent Chord Arc 0000 .0029 .0058 .0087 .0116 .01 15 9° 00' 10 20 30 40 .1564 .1593 .1622 .1650 .1079 .1708 0° 00' 10 20 30 40 50 1° 00 10 20 30 40 50 20 00 10 20 30 40 .000 .0029 .0058 .0087 .0116 .0145 .0175 .0204 0233 .0262 .02991 .0320 .0349 .0378 .0407 0436 .0167 .0494 .0000 .0029 .0058 .0087 .0116 .0145 .0175 .0204 0233 .0262 .0291 .0320 .1569 .1598 .1627 .1656 .1685 .1714 .1743 .1772 .1801 .1830 1859 .1888 10° 00 10 20 30 40 .1736 .1705 .1794 .1822 .1851 .1880 .0175 .0204 0233 .0202 .02.91 .0320 0349 .0378 .0407 .0436 .0465 .0494 .0524 .0553 .0582 .0611 .0610 0469 .1571 .1600 .1629 .1658 .1687 .1716 .1745 .1774 .1804 .1833 .1802 .1891 1920 .1949 .1978 .2007 .2036 .2005 .0319 .0378 .0107 .0436 .0465 .0495 11°00 10 20 30 10 .1908 .1937 1965 .1994 .2022 .2051 .2079 .2108 .2136 .2104 .2193 .2221 3° 00' 10 20 .0523 .0572 .0581 .0610 0X640 .0669 .0524 .05.3 .0582 .0611 12° 00' 10 20 30 40 50 .0659 .0698 .0727 .0750 .0785 .0814 .0843 .0000 .0029 .00.58 .0087 .0116 .0145 .0175 .0204 .0233 .0202 .0291 .0320 .0349 .0378 .0407 .0437 .0400 .0495 .0524 .0553 0582 .0612 .0041 .0670 .0699 .0729 .0758 .0787 ,0816 ,0816 .0875 .0.4 .0934 .0063 .0992 .1022 .1051 .1080 .1110 .1139 .1109 .1198 .1228 .1207 .1287 .1317 .1346 .1376 .1405 .1435 .1465 .1495 .1524 .1554 .1884 .1584 .1614 .1644 .1673 .1703 .1733 .1763 .1793 .1823 .1853 .1883 .1914 .1944 .1974 .2004 .2035 .2005 .2095 .2120 .2156 .2186 .2217 2247 .2278 .2309 2339 .2370 .2401 .2432 .2402 .2493 .2524 .2555 .2580 .2617 .2648 .2679 .2711 .2742 .2773 .2805 .2836 .2867 .2899 .2931 .2962 .2994 .3026 50 13° 00' 10 20 30 40 50 14'00 10) 20 :30 4°00' 10 20 30 40 50 5° 00' 10 20 30 40 50 6° 00' 10 20 30 40 50 70 00 10 20 30 40 40 .0872 .0X1 .0929 .0958 .0987 .1016 1045 .1074 .1103 .1132 .1161 .11.90 .1219 .1248 .)276 .1305 .1334 .1363 .1392 .1421 .1449 .1478 .1507 .1536 .1564 .0698 .0727 .0756 ,0785 .0814 .0843 .0872 .0901 .0931 .0260 .0:189 .1018 .1047 . 1076 .1105 .1134 .1163 .1192 .1221 .1250 1279 .1308 .1337 .1366 .0698 .0727 .0750 .(78.5 ,0814 .081 .0873 .O902 .0931 .0960 .0189 .1018 .1047 .1076 .1105 .1134 .1164 .1193 .1222 .1251 1280 .1309 .1338 .1367 .1396 .1425 .1454 .1484 .1513 .1542 .1971 .1917 .1946 .1975 .2004 .2033 .2062 .2091 .2119 .2148 .2177 .2206 .2235 .2261 .2293 .2322 2351 .2380 .2409 .2437 .2466 .2495 .2524 .2553 .2582 .2611 .2639 .2668 .2097 .2726 2755 .2783 .2812 2841 .2870 2899 .2927 2956 .2985 .3014 .3042 .3071 .3100 .3129 22.50 .2278 .2306 .23:34 .2363 2391 .2419 .2447 .2476 .2504 .2532 .2560 .2588 .2616 .2644 .2672 .2700 .2728 .2756 .2784 .2812 .2840 .2868 .2896 .2924 .2952 .2979 .3007 .3035 .3062 .3090 15°00' 10 20 30 40 50 18° 00' 10 20 30 40 50 17° 00' 10 20 .2094 .2123 .2153 .2182 .2211 .2240 .2209 .2298 .2327 .2356 .2385 .2414 .2443 .2473 .2502 .2531 .2560 .2589 2618 .2647 2676 .2705 .2734 .2763 .2793 .2822 .2851 .2880 .2909 .2938 .2967 .2996 .3025 .3054 .3083 .3113 .3142 8° 00' 10 20 30 40 .1395 .1424 .1453 .1482 .1511 .1540 .1669 .3057 .3089 .3121 .3153 .3185 .3217 .3249 40 50 18° 00' 9° 00' ·iv Quantities Determined by a Given Angle [1 Angle Sino Tangent Chord Arc Angle Sine Tangent Chord Arc .4540 .45656 .4592 .4617 .4043 .4669 4695 .4720 .4746 .4772 .4797 .4823 .4848 .4874 .4899 .4924 .4950 .4975 18° 00' .3090 10 .3118 20 .3145 30 .3173 40 .3201 50 .3228 19° 00' .3256 10 .3283 20 .3311 30 .3338 40 .3365 50 .3393 20° 00' .3420 10 .3448 20 .3475 30 .3502 40 .3529 .3507 21°00' .3584 10 .3611 20 .3638 30 .3665 40 .3692 50 .3719 22° 00' .3746 10 .3773 20 .3800 30 .3827 40 .3854 50 .3881 88° 00'1.39907 10 .3934 20 .3961 30 .3987 40 .4014 50 .4041 24°00' 4067 10 .4094 20 .4120 30 .4147 40 .4173 50 .4200 25° 00' 1..4226 10 .4253 20 .4279 30 .4305 40 .4331 50 .4358 28° 00 .4384 10 .4410 20 .4436 30 .4462 40 .4488 50 .4514 27°00' .4540 .3129 .3157 .3186 .3215 .3244 .3272 .3301 .3330 .3358 .3387 .3116 .3444 .3473 .3502 .3530 .35.59 .3587 .3610 .3645 23673 .3702 .3730 .3759 .3788 .3816 .3815 .3873 .3902 .3930 .39239 .4669 .4697 .4725 .4754 .4782 .4810 .4838 .4867 .4895 .4923 .4951 .4979 .5008 .5036 .5064 .6092 .5120 .5148 .3249 .3281 .3314 .3346 .3378 .3411 .3443 .3476 .3508 .3541 .3574 .3607 .3640 .3673 .3706 .3739 .3772 .3805 .3839 .3872 .3906 .3939 .3973 .4006 .4040 .4074 .4108 .4142 .4176 .4210 .4245 .4279 .4314 .4348 .4383 .4417 .4452 .4487 .4522 .4557 .4592 .4628 .4663 .4699 .4734 .4770 .4806 .4841 .4877 .4913 .4950 .4986 .5022 .5059 .5095 .3142 .3171 .3200 .3229 .3258 .3287 .3316 .3345 .3374 .3103 .3432 .3462 .3491 .3520 .3549 .3578 .3607 .3036 .3665 .3094 .3723 .3752 .3782 .3811 .3840 .3869 .3898 .3927 .3906 .3985 .4014 .4013 .4072 .4102 .4131 .4160 .4189 .4218 .4247 .4276 4305 .4334 .4363 .4392 .4422 .4451 .4480 .4509 .4538 .4567 4596 .4025 4654 4683 .4712 27°00' 10 20 30 40 50 28° 00' 10 20 30 40 50 29° 00' 10 20 30 40 50 30° 00' 10 20 30 40 50 31° 00' 10 20 30 40 0 32° 00' 10 20 30 40 50 38° 00' 10 20) 30 40 50 34°00' 10 20 30 40 .5095 .5132 .6169 .5206 .5243 .5280 .5317 .5354 .5392 .5430 .5407 .5505 .5543 .5581 .7019 .5658 .56.96 .5735 .5774 .5812 .5851 .5890 .5930 .5.169 .6009 .6048 .6088 .6128 .6168 .6208 .62419 .6289 .6330 .6371 .6412 .6453 .5000 .5025 .5050 .5075 .5100 .5125 .5150 .5175 .5200 .5225 .5250 .5275 .5290 .5324 .5348 .5373 .5398 .5422 .4712 .4741 .4771 .4800 .4829 .4858 .4887 .4916 .4945 .4974 .5003 .5032 .5061 .5091 .5120 .5149 .5178 .7207 .5236 .5265 .5224 .5323 .5352 .5381 .5411 .5440 ,5469 .7498 .5527 .5556 .5585 .5614 .5643 .5672 .5701 .5730 .5760 .5789 .5818 .5847 .8876 .5905 .5934 .5963 .8992 .6021 .6050 .6080 .6109 .6138 .6167 .6196 .6225 .6254 .6283 .3987 .4016 .4041 .4073 .4101 .4130 .4158 .4187 .4215 .4244 .4272 4300 .4329 4357 .4386 .4414 .4442 .4471 .4499 .4527 .4556 .4584 .4612 .4641 .5446 ..5171 .3495 .5519 .5544 .5568 .5592 .5616 .5640 .5644 .5688 .5712 .8736 .5760 .5783 .5807 .5831 .5854 .8878 .5176 .5204 .5233 .5261 .5289 .5317 .5345 .5373 .15401 .5429 .54157 .6485 .5513 .5541 .5569 .5597 .5625 .5652 .5680 .5708 .5736 .5764 .5792 .5820 .5847 .5875 .5903 .5931 .5989 .5986 .6014 .6042 .6070 .6097 .6125 .6153 .6180 .6494 .6536 .6577 .6619 .6661 .6703 .6745 .6787 .6830 .6873 .6916 .6959 .7002 .7046 .7089 .7133 .7177 .7221 .7265 50 85° 00' 10 20 30 40 50 36° 00' .4669 ·1] Quantities Determined by a Given Angle Angle Sine Tangent Chord Arc Angle Sino Tangent Chord Arc 45° 00' 10 20 30 40 46°00' 10 20 30 40 50 47°00' 10 20 30 40 50 48°00' 10 20 30 40 36° 00' .5878 10 .5901 20 .5925 30 .5948 40 .5972 50 .5995 37°00' .6018 10 .6041 20 .6065 30 .6088 40 .6111 50 .6134 38°00' .6157 10 .0180 20 .0202 30 .6225 40 .6248 .6271 39° 00' .0293 10 .6316 20 .6338 30 .6361 40 .6383 50 .6406 40000 .6428 10 .6450 20 .6472 30 .6494 40 .6517 50 .6539 41°00' .6561 10 .6583 20 .6604 30 .6626 40 .6648 50 .6670 42° 00' .6691 10 .6713 20 .6734 30 .6756 40 .6777 50 .6799 43°00' .6820 10 .6841 20 .6862 30 .6884 40 .6.905 .6926 44° 00' .0947 10 .6967 20 .6988 30 .7009 40 .7030 50 .7050 45° 00'1.7071 .7265 .6180 .7310 .6208 .7355 .6236 .7400 .6263 .7445 .6291 .7490 .6318 .7536 .6346 .7581 .6374 .7627 .6401 .7673 .6429 .7720 .6456 .7766 .6184 .7818 .6511 .7860 .6539 .7907 .6566 .7954 .6594 .8002 .6621 .8050 .6649 .8098 .6076 .8146 .6701 .8195 .6731 .8943 .6758 .82912 .6786 .8342 .6813 .8391 .6840 .8441 .6868 .8491 .6895 .8541 .6922 .8591 .69570 .8042 .6977 .8693 .7004 .8744 .7031 .8796 .70599 .8847 .7086 .8899 .7113 .89524140 .9004 .7167 .9057 .7195 .9110 .7222 .9163 .7249 .9217 .7276 .9271 .7303 .9325 .7330 .9380 .7357 .9435 .7384 .9490 .7411 .9745 .7438 .9601 .74655 .9657 .7492 .9713 .7519 .9770 .7546 .9827 .7573 .9884 .7600 .9942 7627 .6283 .6312 .6341 .6370 .6400 .6429 .6458 .6487 .6516 .6545 .6574 .6003 .6032 .6661 .609 .6720 .6749 .6778 .6807 .6836 .6865 .6894 .6923 .6952 .6981 .7010 .7039 .7069 .7098 .7127 .71067 .7185 .7214 .7243 .7272 .7301 .7330 .7359 .7389 .7418 .7447 .7476 .7505 .7534 .7563 .7592 .7621 .7690 49° 00' 10 20 30 40 50 50°00' 10 20 30 40 50 51° 00' 10 20 30 40 50 52000 10 20 30 40 50 63° 00' 10 20 30 40 50 54°00' .7071 1.0000 7654 .7092 1.0058 .7681 .7112 1.0117 .7707 .7133 1.0176 .7734 .7153 1.0235 .7761 .71731.0295 .7788 .7193 1.0355 .7815 .7214 11.0416 .7841 .7234 1.0477 .7868 .7254 1.0538 .7895 .7274 1.0599.7922 .7294 1.0661 .7948 .7314 11.0724 .7975 .7333 1.0786 .8002 .7353 1.0850 .8028 .7373 1.0913 .8055 .7392 1.0977 .8082 .7412 1.1041 .8108 .7431 1.1106 .8135 .7451 1.1171 .8161 .7470 1.1237 .8188 .7490 1.1303 .8214 .7509 1.1369 .8241 .7528 1.1436 .8267 .7547 1.1504 .8294 .7566 1.1571 .8320 .7585 1.1640 .8347 .7604 1.1708 .8373 .7623 1.1778 .8400 .7642 1.1847 .8426 .7660 1.1918 .8452 .7679 1.1988 .8479 .7698 1.2059 .8505 .7716 1.2131 .8531 .7735 1.2203 .8558 .7753 1.2276 .8584 .7771 1.2349 .8610 .7790 1.2423 .8636 .7808 1.24197 .8663 .7826 1.2572 .8689 .7844 1.2647 .8715 .7862 1.2723 .8741 .7880 1.2799 .8767 .7898 1.2876 .8794 .7916 1.2954 .8820 .7934 1.3032 .8846 .7951 1.3111 .8872 .7969 1.3190 .8898 .7986 1.3270 .8924 .8004 1.3351 .8950 .8021 1.3432 .8976 .8039 1.3514 .9002 .8056 1.3597 .9028 .8073 1.3680 9054 .8090 1.3764 .9080 .7854 .7883 .7912 .7941 .7970 .7999 .8029 .8058 .8087 .8110 .8145 .8174 .8203 .8232 .8261 .8290 .8319 .8348 .8378 .8407 .8436 .8465 .8494 .8523 .8552 .8581 .8610 .8639 .8668 .8698 .8727 .8756 .8785 .8814 .8843 .8872 .8901 .8930 .8959 .8988 .9018 .9047 .9076 .9105 .9134 .9163 .9192 .9221 .9250 .9279 .9308 .9338 .9367 .9396 11: .7079 .7709 .7738 .7767 .7796 .7825 .7854 1.0000 .7654 .9425 ·vi Quantities Determined by a Given Angle [I Angle Sine Tangent Chord Arc Angle Sine Tangent Chord Arc 10 630 00 10 20 30 40 50 64°00' 10 20 30 40 50 65° 00' 10 20 30 40 0 66°00' 10 20) 30 40 50 67°00' 10 64° 00' .8090 1.3764 .9080 .9425 .8107 1.3848 .9106 .9454 20 .8124 1.3934 .9132 .9483 30 .8141 1.4019 .9157 .9512 40 .8158 1.4106 .9183 .9541 50 .8175 1.4193 .9209 .9570 56°00' .8192 1.4281 .9235 .9599 10 .8208 1.4370 .9261 .8628 20 .8225 1.4460 .9287 .9657 30 .8241 1.4550 .9312 .8687 40 .8258 1.4641 .9338 .9716 50 .8274 1.4733 .9364 .9745 58°00' .8290 1.4826 .9389 .9774 10 .8307 1.4919 .9415 .9803 20 .8323 1.5013 .9171 .9832 30 .8339 1.5108 .2466 .9861 40 .83551.5204 .2492 .980 50 .8371 1.5301 .9518 .9919 57°00' .8387 1.5399 .9543 .9948 10 .8403 1.0497 .9569 .9977 20 .8418 1.5597 .95.24 1.0007 30 .8434 1.5697 .4620 1.0036 40 .8450 1.5798 .9645 1.0065 50 .8465 1.5XX) .871 1.00 58°00' .8480 1.6003 .3696 1.0123 10 .81.5% 1.6107 .9722 1.0152 20 .8511 1.6212 .9747 1.0181 30 .8526 1.6313 .9772 1.0210) 40 .8542 1.0426 .979081.0239 50 .8557 1.0534 .4823 1.0268 59°00'.8572 1.6613 .9818 1.0297 10 .8587 1.6753 .9874 1.0327 20) .86501 1.6864 .9899 1.0356 30 .8616 1.6977 .992+ |1.0385 40 .8631 1.7090 .9950 1.0414 .8646 1.7205 .9975 1.0143 60000 1.7321 1.0000 1.0472 10 .84575 1.7437 1.0025 1.0501 20 .8689 1.755€ 1.0050 1.0530 30 .8704 1.7675 1.0075 1.0559 40 .8718 1.77% 1.0101 1.0588 50 .8732 1.7917 1.0126 1.0617 61°00' .8746 1.80401.0151 1.0647 10 .8760 1.8165 1.0176 1.0676 20 .8774 1.8291 1.0201 1.0705 30 .8788 1.8418 1.0226 1.0734 40 .8802 1.8546 1.0251 1.0763 50 .8816 | 1.8076 1.0276 1.0792 62° 00' .8829 1.8807 1.0301 1.0821 10 .8843 1.8940 | 1.0326 1.0850 20 .8857 1.9074 1.0351 1.0879 30 .8870 1.9210 1.0375 1.0908 40 .8884 1.9347 | 1.0400 I 1.0937 50 .8897 1.9486 | 1.0425 1.0966 63° 00'.8910 1.9626 | 1.0450 1.0996 30) 40 50 68° 00' 10 20 30 .8910 1.9626 1.0450 1.0996 .8923 1.9768 1.0475 1.1025 .8936 1.9912 | 1.0500 1.1054 .8949 2.0057 1.0524 1.1083 .89%82 2.0204 1.0549 1.1112 .8975 2.0353 (1.0574 / 1.1141 .8988 2.0003 1.0598 1.1170 .X01 2.0055 1.0623 1.1199 .9013 2.0609 1.0648 1.1228 SM)26 2.0.265 1.0672 1.1257 .9038 2.1123 1.0697 1.1286 ..051 2.1283 1.0721 1.1316 9063 2.1445 1.0746 1.1345 .X075 2.1609 1.0771 1.1374 .:M88 2.1775 1.0795 1.1403 .9100 2.143 1.0820 1.1432 .:0112 2.2113 1.0844 1.1461 .9124 2.2286 1.0868 1.1490 .9135 2.2400 1.0893 1.1519 .9147 2.2637 | 1.09171.1548 .9159 2.2817 1.09421.1577 .9171 2.2998 1.026 1.1606 .9182 2.3183 1.090 1.1636 .91.94 2.3309 | 1.1014 1.1665 .9205 2.3559 1.1039 1.1694 .9216 2.3750 1.1063/1.1723 .9228 2.325 1.1087 1.1752 .92:39) 2.11421.11111.1781 .9250 2.4342 1.1136 | 1.1810 .9261 2.4515 1.1100 1.1839 .9272 2.4751 1.1184 1.1868 .9:283 2.480 | 1.1208 | 1.1897 .9293 2.51721.1232 1.1926 9:304 2.5386 1.1056 1.1956 .9315 2.5005 1.1280 1.1985 .9325 2.5826 1.1304 1.2014 .9336 2.6051 1.1328 1.2043 .9346 2.6279 1.1352 1.2072 .9356 2.6511 1.1376 1.2101 9:367 2.0746 1.1400 1.2130 .9377 2.6985 1.1424 1.2159 .9387 2.7228 1.1448 1.2188 .9397 2.7475 1.1472 1.2217 .9107 2.7725 1.1495 1.2246 .9417 2.7980 1.1519 1.2275 .9426 2.8239 1.1543 1.2305 .9436 2.8502 1.1567 1.2334 .9446 2.8770 1.1590 1.2363 .9455 2.9042 | 1.1614 1.2392 .9465 2.93191.1638 1.2421 .9474 2.9600 1.1661 1.2450 .9483 2.9887 | 1.1087 1.2479 .9492 3.0178 1.1709 1.2508 .9502 3.0475 1.1732 1.2037 .9511 3.0777 1.1756 1.2666 40 50 690 00 10) 20 30 40 50 70° 00' 10 20 30 40 50 71° 00' 10 20 30 40 50 720 00 ·'] Quantities Determined by a Given Angle vii Angle Sine Tangent Chord Arc Angle Sine Tangent Chord Arc 10 72°00' .9511 3.07771.1756 1.2566 10 .9520 3.1081 1.1779 1.2595 20 .9528 3.1397 1.18031.2627 30 .9537 3.1716 1.1826 1.2654 40 .9546 3.2041 1.1850 | 1.2683 .9555 3.2371 1.1873 1.2712 73° 00' .9563 3.2709 1.1896 1.2741 .9572 3.3052 1.1920 1.2770 0 .9580 3.:3402 1.1943 1.279) 30 .9588 3.37591.1966 1.2828 40 .95% 3.4124 1.1900 1.2857 50 X305 3.4495 1.20131.2886 74°00 .813 3.4874 1.2036 1.2915 10 .*:21 3.5261 1.2000 1.2145 20 %628 3.5650 1.20831.2974 30 3336 3.60.591.2106 1.3003 40 .2014 3.6470 1.2129 1.3032 50 .852 3.6891 1.2152 1.3061 75°00' .9659 3.7321 1.2175 1.30,70 10 .86657 3.7760 1.2198 1.3119 20 .9674 3.8208 1.2221 1.3148 30 .981 3.86071.2244 1.3177 40 ./89 3.91361.2207 1.3200 50 .9630 3.96171.2290 1.3933 76°00' .9703 4.0108 1.2313 | 1.3205 10 .9710 4.0611 1.2330 1.32014 20 .9717 4.1126 1.2359 | 1.3323 30 .9724 4.1053 1.2:3821.3352 40 .97:30 4.2193 1.2405 1.3381 .9737 1.2747 1.2428 1.3410 77°00' .9744 | 4.3315 1.2150 1.3139 10 .9750 4.3897 1.2473 1.3408 20 .9757 4.4494 1.2436 1.3497 30 .9763 4.5107 1.2518 1.3526 40 .9709 4.5736 1/2541 1.3505 50 .9775 4.6382 1.2564 1.3584 78° 00 .9781 4.7046 1.2586 1.3614 10 9787 4.77291.2609 1.3643 20 .9793 4.8430 | 1.2632 1.3672 .9799 4.91521.26541.3701 40 .9805 4.9894 1.2677 1.3730 50 .9811 5.0658 1.2699 1.3759 790 00 .9816 5.1446 1.2722 1.3788 10 .9822 5.2257 1.2744 1.3817 .9827 5.3093 1.2766 1.3846 30 :9833 5.3955 1.2789 1.3875 40 .9838 5.4815 1.2811 |1.3904 50 .9843 5.5764 1.2833 1.3934 80° 00' .9848 5.6713 1.2856 1.3963 10 .9853 5.7694 1.2878 1.3992 20 .9858 5.8708 1.2900 1.4021 30 .9863 5.9758 1.2922 1.4050 40 .9868 6.0844 1.2345 1.4079 50 .9872 6.1970 1.2967 1.4108 81°00' .9877 6.3138 1.2989 1.4137 81° 00' .98776.3138 1.2989 1.4137 10 .98816.4318 1.3011 1.4166 20 .9886 6.5606 1.3033 1.4195 30 .980 6.6912 1.:3055 1.4224 40 .9894 6.8269 | 1.3077 1.4254 50 .9899 6.9682 1.3099 1.4283 82° 00' .9903 7.1154 1.3121 1.4312 10 .9907 | 7.2687 | 1.3143 1.4341 20 .9911 | 7.4287 1.3165 1.4370 30 .9914 7.5958 1.3187 1.4399 40 .9918 7.7704 1.32091.4428 .9922 7.9530 1.3231|1.4457 83° 00' .9927 8.1443 1.3252 1.4486 10 .9929 8.3450 1.3274 1.1515 20 .99328.5555 1.32% 1.4514 30 .99368.77691.3318 1.4573 40 .99399.0098 1.3339 1.4603 30 .9942 9.2553 1.3361 1.4632 84° 00' .9945 9.5144 1.3383 1.4661 10 .99489.78821.3104 1.4690 20 .9951 10.078 1.3426 1.4719 30 .9954 10.385 1.3447 | 1.4748 .9957 10.712 1 1.3169 1.4777 50 .9959 11.0591.3490 1.4806 850 000 .9262 | 11.130 1.35121.4835 10 .967 11.826 1.3533 1.4864 20 .987 12.251 1.3555 1.4893 30 .9769 12.706 1.35761.4923 .9971 13.1971.3597 | 1.4952 50 .9974 13.727 1.3619 1.4981 86° 00' .997614.301 1.3640 1.5010 10 .9978 14.924 1.3661 1.5039 20 .9980 15.605 1.36821.5068 30 .9981 16.350 1.3704 1.5097 40 .9983 17.169 1.3725 1.5126 50 .9985 18.075 1.3746 1.5155 87°00' .9986 19.081 1.3767 1.5184 10 .9988 20.206 1.3788 1.5213 20 .9989 21.470 1.38091.5243 30 .9990 | 22.904 1.3830 1.5272 40 .9992 24.542 1.3851 1.5301 50 .9993 26.432 1.3872 1.5330 88° 00 .9994 28.636 1.3893 1.5359 10 .9995 31.242 1.3914 1.5388 20 .99x634.368 1.3935 1.5417 30 .9997 38.188 1.3956 1.5446 40 .9997 42.964 1.3977 1.5475 50 .9998 49.104 1.3997 1.5504 89°00' .9998 57.290 1.4018 1.5033 10 .9999 68.750 1.4039 1.5563 .9999 85.940 1.4060 1.5592 30 1.0000 114.59 1.4080 1.5621 40 1.0000 171.89 1.4101 1.5650 50 1.0000 343.77 1.4122 1.5679 90° 00' 1.000.0 1.4142 1.5708 40 30 20 ·TABLE II -- POWERS AND ROOTS EXPLANATION OF TABLE II 1. Squares and Cubes. The squares of numbers between 1.00 and 10.00 at intervals of .01 are given in column headed n. To find the square of any other number, divide (or mul. tiply) the given number by 10 to reduce it to a number between 1 and 10; find the square of this last nuinber; multiply (or divide) the square thus found by 10 twice as many times as you did the given number. The cube is given in the column headed n?. To find the cube of any number not between 1 and 10, first reduce that number to a number between 1 and 10 by dividing (or multiplying) by a power of 10. Multiply (or divide) the result found by three times as high a power of 10 as was used to reduce the given number, 2. Square Roots. The square roots of numbers between 1 and 10 are found in the column headed vn. The square roots of nuinbers between 10 and 100 may be found in the column headed V10 n. The square roots of numbers between 100 and 1000 may be found in the column headed vn by multiplying the given root by 10, since V100 n = 10 Vm. Other square roots may be found in a similar manner. 3 3. Cube Roots. The column headed : Yn gives cube roots of numbers between 1 and 10; 10n gives cube roots of numbers between 10 and 100; V100n gives cube roots of numbers between 100 and 1000. To find the cube root of a number between 1000 and 10000, take 10 times the value found in the column headed ºn, since V1000n = 10 V. Other cube roots may be found similarly. viii ·II] Table 11 Powers and Roots ix n n2 vn V10 n ns in 10 n 100 n 1.00 1.0000 1.00000 3.10228 1.00000 1.00000 2.1.54.13 4.64159 1.01 1.02 1.03 1.0201 1.0404 1.0609 1.004:3) 1.0099.7 1.01489 3.1780.7 3.19.374 3.2008 1.030:30 1.06121 1.09273 1.003:32 1.00662 1.00990 2.16109 2.16870 2.17577 4.67701 4.67233 4.68755 1.04 1.05 1.06 1.0816 1.1025 1.1230 1.01980) 1.024170 1.02956 3.224.00 3.24037 3.25576 1.12486 1.1.762 1.19102 1.01316 1.01640 1.01961 2.18279 2.18976 2.1X69 4.70267 4.71769 4.73202 1.07 1.08 1.09 1.1449 1.1664 1.1881 1.03441 1.03923 1.04403 3.27109 3.28034 3.30151 1.22504 1.25971 1.29503 1.02281 1.02599 1.02914 2.20358 2.21042 2.21722 4.74746 4.76220 4.77686 1.10 1.2100 1.04881 3.31002 1.33100 1.03228 2.22398 4.79112 1.11 1.12 1.13 1.2:321 1.2514 1.2769 3.33107 3.31661 3.30155 1.30763 1.40403 1.44290 1.03540 1.03850 1.041.58 2.23070 2.23788 2.24402 4.80590 4.82028 4.83459 1.14 1.15 1.16 1.2996 1.3225 1.3456 1.05357 1.05830 1.00301 1.06771 1.07238 1.07703 1.08167 1.08028 1.0787 3.37039 3.30110 3.40588 1.48154 1.72098 1.560W 1.0401 1.04709 1.0507 2.2002 2.25718 2.26370 4.84881 4.80294 4.87700 1.17 1.18 1.19 1.3089 1.3924 1.4161 3.42053 3.43711 3.41964 1.60101 1.01303 1.68516 1.05373 1.05072 1.03970) 2.27019 2.27601 2.28305 4.89097 4.90487 4.91868 1.20 1.4400 1.09515 3.40410 1.72800 1.00266 2.28943 4.93242 1.21 1.22 1.23 1.4611 1.4884 1.5129 1.10000 1.1015+ 1.10205 3.47871 3.19285 3.50714 1.77156 1.81585 1.80087 1.00560 1.00853 1.07141 2.29077 2.3208 2.30835 4.94609 4.95968 4.97319 1.24 1.25 1.26 1.5376 1.5625 1.5876 1.11355 1.11803 1.12250) 3.121:30 3.535.73 3.5 1905 1.90662 1.95312 2.00038 1.07434 1.07722 1.08008 2.31459 2.32079 2.32097 4.98663 5.00000 5.01330 1.27 1.28 1.29 1.6129 1.0384 1.6011 1.12691 1.13137 1.13078 3.56371 3,57771 3.5.166 2.048:38 2.09715 2.1.1669 1.08293 1.08577 1.08859 2.33311 2.33921 2.34529 5.02653 0.03768 5.05277 1.30 1.0200 1.14018 3.60955 2.19700 1.09139 2.35133 5.06580 1.31 1.32 1.33 1.7161 1.712+ 1.7689 1.14455 1.14801 1.15326 3.61939 3.033318 3.61692 2.21809 2.29997 2.35204 1.09118 1.096 1.09972 2.37735 2.30333 2.300128 1.34 1.35 1.345 1.7956 1.8225 1.847 1.15758 1.16190 1.10619 3.66060 3.67423 3.68782 2.40610 2.40038 2.51016 2.37521 2.38110) 2.38097 1.10247 1.10521 1.10793 1.11061 1.11: 34 1.11602 5.07875 5.09164 5.10447 5.11723 5.12993 5.142561 5.15514 5.16765 5.18010 1.37 1.38 1.39 1.8769 1.2014 1.9:321 1.17047 1.17473 1.17898 3.70135 3.71481 3.72827 2.57135 2.62807 2.68562 2.39280 2.39801 2.404:39 1.40 1.200 1.18322 3.74166 2.74400 5.19249 1.41 1.42 1.43 1.9881 2.0164 2.0449 1.18713 1.19161 1.19583 2.80322 2.80329 2.92421 1.11869 1.12135 1.12399 1.12662 2.41014 2.41587 2.42156 2.42724 5.20483 5.21710 5.22932 1.44 1.45 1.46 2.0736 2.1025 2.1316 1.20000 1.20416 1.20830 3.75500 3.70829 3.78153 3.79473 3.80789 3.82099 3.83406 3.84708 3.86005 2.98598 3.01802 3.11214 3.17652 3.24179 3.30705 1.121124 1.13185 1.13145 1.13703 1.13.30 1.14216 2.43288 2.43850 2.41409 2.44X60 2.45520 2.40072 5.24148 5.2.5.359 5.26564 5.27763 5.28957 5.30146 1.47 1.48 1.49 2.1609 2.1904 2.2201 1.21244 1.21055 1.22066 ·Powers and Roots n n2 vn V10n V10 n V100 n 1.60 2.2500 1.22174 3.87298 3.37500 1.14471 2.46621 1.51 1.02 1.53 2.2801 2.3104 2.3109 1.22882 1.23288 1.23093 3.44295 3.51181 3.58158 1.14725 1.14978 1.15230 2.47168 2.47712 2.48255 5.31329 5.32507 5.33680 5.34848 3.88087 3.89872 3.91152 3.92428 3.93700 3.94968 1.54 1.55 1.56 2.3716 2.4025 2.4336 1.21097 1.24499 1.24900 3.65226 3.72388 3.79642 1.15480 1.15729 1.18978 2.48794 2.49332 2.49867 5.36011 5.37169 5.38321 6.39469 5.40612 5.41750 1.57 1.58 1.59 2.4649 2.4964 2.5281 1.25300 1.25098 1.20095 3.96232 3.97492 3.98748 3.86989 3.94431 4.01268 1.16225 1.16471 1.10717 2.50399 2.509:30 2.51458 1.60 2.5600 1.26491 4.00000) 4.09600 1.1641 2.51984 5.42884 1.61 1.62 1.63 2.5921 2.6244 2.6569 4.17328 4.25153 4.33075 2.52508 2.53030 2.53549 5.44012 5.45136 5.46256 1.20886 1.27279 1.27671 1.28062 1.28152 1.28841 4.01248 4.02492 4.03733 4.0429 4.06202 4.07431 1.17201 1.17446 1.17687 1.17927 1.18167 1.18105 2.6896 2.7225 2.7556 4.41091 4.49212 4.57430 2.54067 2.54582 2.55095 5.47370 5.48481 5.49586 1.64 1.65 1.66. 1.67 1.68 1.69 2.7889 2.8224 2.8561 2.8.900 1.29228 1.29615 1.30000 1.30384 4.08656 4.09878 4.11096 4.65746 4.74163 4.82681 4.91300 1.18612 1.18878 1.19114 2.55607 2.56116 2.56023 5.30688 5.51785 5.62877 4.12311 1.19348 2.57128 5.53966 1.70 1.71 1.72 1.73 1.74 1.75 1.76 2.9211 2.9584 2.9929 1.19582 1.19815 1.20016 5.56060 5.56130 5.57205 1.30767 1.31149 1.31529 1.31909 1.32288 1.32665 4.13521 4.11729 4.15933 4.17133 4.18330 4.19524 4.20714 4.21200 4.23084 3.0276 3.0025 3.0976 5.00021 5.08815 5.17772 5.26802 5.35938 5.45178 5.54523 5.63975 5.735.34 2.57631 2.08133 2.58632 2.89129 2.52625 2.60118 5.58277 5.59344 5.60408 1.20277 1.20.507 1.20736 1.20964 1.21192 1.21418 1.77 1.78 1.79 3.1329 3.1684 3.2011 1.33041 1.33417 1.33791 2.00610 2.61100 2.61588 5.61467 5.62023 5.63574 1.80 3.2400 1.34104 4.24201 5.83200 1.21644 2.62074 5.64622 3.2761 3.3124 3.3489 1.21809 1.22093 1.22316 2.62559 2.03041 2.63522 1.81 1.82 1.83 1.84 1.85 1.86 1.34536 1.34907 1.35277 1.35647 1.36015 1.36382 1.36748 1.37113 1.37477 4.23441 4.26615 4.27785 4.28952 4.30116 4.31277 5.92974 6.02857 6.12849 6.22950 6.33102 6.43486 3.3856 3.4225 3.45%6 1.22539 1.22760 1.22981 2.64001 2.64479 2.64954 1.87 1.88 1.89 3.4969 3.5344 3.5721 4.32435 4.335500 4.34741 6.53920 6.64467 6.75127 6.85900 1.23201 1.23420 1.23639 2.65428 2.05901 2.60371 3.6100 1.37840 4.35890 1.23856 1.90 1.91 1.92 1.93 5.65665 5.66705 5.67741 5.68773 5.69802 5.70827 5.71848 5.72865 5.73879 5.74890 5.70897 5.76900 6.77900 5.78896 5.79889 5.80879 5.81863 5.82848 0.83827 3.6481 3.6864 3.7249 1.38203 1.38561 1.38924 4.37035 4.38178 4.39318 6.96787 7.07789 7.18906 1.24073 1.24289 1.24505 3.7636 3.8026 3.8416 3.8809 3.9204 3.9601 1.985 1.96 1.39284 1.39642 1.40000 1.40357 1.40712 1.41067 4.40454 4.41588 4.42719 4.43847 4.44972 4.46094 2.66840 2.67307 2.67773 2.68237 2.68700 2.69101 2.69620 2.70078 2.70534 2.70989 7.30138 7.41488 7.52954 7.64537 7.76239 7.88060 1.24719 1.24933 1.25146 1.20359 1.25571 1.25782 1.97 1.08 1.99 ·11) Powers and Roots xi vn V10 n in $10 n 100 n 2.00 4.0000 1.41421 4.47214 8.00000 1.25992 5.84804 2.01 2.02 2.03 4.0401 4.0801 4.1209 1.41774 1.42127 1.42478 4 48330 4.4444 4.50555 8.12000 8.24241 8.36543 1.26202 1.26111 1.26619 2.71442 2.71893 2.72344 2.72792 5.85777 5.80746 5.87713 2.04 2.05 2.06 4.1016 4.2025 4.2436 1.42829 1.43178 1.43527 4.51664 4.52709 4.53872 8.48966 8.61512 8.74182 2.73239 2.73685 2.74129 5.88677 5.87637 5.90594 1.20827 1.27033 1.27240 1.27445 1.27000 1.27854 4.2849 4.3264 4.3481 4.4100 1.43875 1.44222 1.44568 4.54973 4.56070 4.57165 8.80974 8.99891 9.12933 2.74572 2.75014 2.75454 5.91548 5.92499 5.93447 1.44917 4.58258 9 26100 1.28058 2.75892 5.91302 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 4.4521 4.4944 4.5369 1.45258 1.40002 1.45945 9.39393 9.52813 9.66300 1.28261 1.28463 1.28605 2.76330 2.70766 2.77200 5.95334 5.96273 5.97209 4.59347 4.60435 4.61519 4.02001 4.63681 4.64758 4.6796 4.6225 4.6656 1.40287 1.40029 1.46969 9.80034 9.93838 10.0777 1.28860 1.22066 1.29266 2.77633 2.78065 2.784955 5.98142 5.99073 6.00000 2.17 2.18 2.19 4.7089 4.7524 4.7%1 1.47309 1.47618 1.47986 4.65833 4.06.205 4.67974 10.2183 10.3602 10.5033 1.29465 1.24664 1.29802 2.78924 2.79352 2.79779 6.00925 6.01846 6.02765 2.20 4.8400 1.48324 4.69042 10.6480 1.30089 2.80204 6.03081 2.21 2.22 2.23 4.8841 4.928+ 4.9729 1.18661 1.48997 1.49332 10.7939 10.94.10 11.0896 2.80628 2.81050 2.81472 6.04594 6.05505 6.06413 4.70106 4.71109 4.72229 4.73286 4.74342 4.70395 1.30256 1.30452 1.30648 1.30813 1.31037 1.31231 2.24 2.25 2.26 5.0176 5.0025 5.1076 1.49666 1.50000 1.50333 11.2394 11.3906 11.5432 2.81892 2.82311 2.82728 6.07318 6.08220 6.09120 2.27 2.28 2.29 5.1529 5.1984 5.2441 1.50605 1.50997 1.51327 4.76445 4.77493 4.78539 11.6971 11.8524 12.000 1.31424 1.31617 1.31809 2.83145 2.83560 2.83974 6.10017 6.10911 6.11803 2.30 5.2000 1.51658 4.79.583 12.1070 1.32001 2.84387 6.12613 2.31 2.32 2.33 5.3361 5.3824 5.4289 1.51987 1.52315 1.52043 4.80025 4.81667 4.82701 12.3264 12.4872 12.6493 1.32192 1.32382 1.32572 2.84798 2.85209 2.85618 6.13579 6.14463 6.15345 2.34 2.35 2.36 5.4756 5.5225 5.5696 1.52971 1.53297 1.03623 4.83735 4.84708 4.85798 12.8129 12.9779 13.1443 1.32761 1.32950 1.33139 2.86026 2.86433 2.86838 6.16224 6.17101 6.17975 2.37 2.38 2.39 2.40 5.6169 5.0647 5.7121 1.53948 1.54272 1.54546 4.86826 4.87852 4.88876 13.3121 13.4813 13.6519 1.33326 1.33514 1.33700 2.87243 2.87646 2.88049 6.18846 6.19715 6.20582 6.21447 5.7600) 1.54919 4.89898 13.8240 1.33887 2.88-150 2.41 2.42 2.43 5.8081 5.8504 5.9049 1.55242 1.55503 1.55885 13.9975 14.1725 14.3489 1.31072 1.34277 1.34442 2.88850 2.89249 2.8%47 6.22308 6.23168 6.24020 4.90918 4.91935 4.92950 4.93964 4.94975 4.95984 2.44 2.45 2.46 5.9536 6.0025 6.0516 14.5268 14.7061 14.8869 1.34626 1.34810 1.34993 2.90044 2.90439 2.90834 6.24880 6.25732 6.26583 1.86205 1.56525 1.86844 1.57162 1.57480 1.67797 2.47 2.48 2.49 6.1009 6.1504 6.2001 4.96991 4.97996 4.98999 15.0692 15.2530 15.4382 1.35176 1.35358 1.35540 2.91227 2.91620 2.92011 6.27431 6.28276 6.29119 + ·Powers and Roots (II n n vn V10 n ns yn 310 n 100 n 6.2500 1.58114 5.000.00 15.6250 1.30721 2.92402 6.29961 2.50 2.51 2.52 2.53 6.3001 6.3504 6.4009 1.68430 1.58745 1.59060 5.00999 5.019020 5.02991 15.8133 16.0030 16.1943 1.37902 1.30082 1.30262 2.92791 2.93179 2.93507 6.30799 6.31636 6.32470 2.51 2.55 2.50 6.4510 6.5020 6.5536 1.59374 1.5%87 1.60000 5.03984 5.04975 5.05%64 16.3871 16.0814 16.7772 1.36441 1.30620 1.3198 2.93953 2.94338 2.44723 0.33303 6.34133 6.34960 2.57 2.58 2.59 6.6049 0.0564 6.7081 1.00312 1.60*24 1.60935 1.61245 5.06952 5.07937 5.08920 16.9746 17.1735 17.3740 17.5700 1.36976 1.37153 1.37330 2.95106 2.93488 2.99869 6.35786 0.36610 6.37431 2.60 0.7000 5.09:02 1.37507 2.90250 0.382570 2.61 2.62 2.63 5.10882 5.11859 5.12835 17.7796 17.9847 18.1914 1.37683 1.37859 1.38034 2.96629 2.97007 2.97385 6.32068 6.39883 6.40696 6.8121 6.8014 6.9169 6.9690 7.0225 7.0756 7.1289 7.1824 7.2:301 1.61555 1.61864 1.62173 1.62181 1.02788 1.63095 2.64 2.65 2.66 6.1:3809 5.14782 5.15752 18.3997 18.009% 18.8211 1.38208 1.38383 1.38557 2.97761 2.98137 2.98511 6.41707 6.42316 6.43123 1 6.43928 6.44731 6.45531 2.67 2.68 2.69 2.70 1.03401 1.03707 1.01012 5.10720 5.17087 5.18032 19.0342 19.2488 19.4651 1.38730) 1.38903 1.39076 2.98885 2.99257 2.9629 7.2X0 1.64317 5.1.2015 19.08:30 1.39218 3.00000 6.46330 2.71 2.72 2.73 1.01021 1.01921 1.05227 5.20577 8.21.536 0.22491 19.:X25 20.1236 20.3464 1.3.119 1.39591 1.39761 3.00370 3.00739 3.01107 6.47127 6.47922 6.48715 7.3141 7.3981 7.4529 7.5076 7.5625 7.6176 7.6729 7.7284 7.7811 5.2:$150 5.24104 5.25357 20.5708 20.79769 21.0210 1.39932 1.-10102 1.40272 3.01474 3.01811 3.02200 2.74 2.75 2.76 2.77 2.78 2.79 6.49507 6.502980 6.51083 1.05529 1.05831 1.60132 1.66433 1.66733 1.67033 5.26308 0.27257 5.28205 21.2539 21.4550 21.7176 1.40441 1.40010 1.40778 3.02570 3.02934 3.03297 6.51808 1 6.52052 6.53134 6.51213 2.80 7.8400 5.29150 21.9520 1.40946 3.03059 2.81 2.82 2.83 7.8901 7.9524 8.0089 1.67332 1.07031 1.67929 1.68226 5.30094 0.31037 5.31977 22.1880 22.4208 22.6052 1.41114 1.41281 1.41448 3.04020 3.0-380 3.04740 6.54991 6.55767 6.56541 8.0056 8.1225 8.1796 1.08523 1.08819 1.69115 5.32917 5.33854 5.34790 22.2003 23.1491 23.3937 1.41614 1.41780 1.41946 3.05098 3.05150 3.05813 6.57314 6.58084 6.58853 2.84 2.80 2.86 2.87 2.88 2.89 8.2369 8.224+ 8.3521 1.09411 1.09700 1.70000 1.7024 5.35724 5.30656 5.37587 23.0399 23.8879 24.1376 1.42111 1.42276 1.42440 3.06169 3.06524 3.06878 6.59620 6.60385 6.61149 2.90 8.4100 5.38516 24.38910 1.42604 3.07232 6.61911 2.91 2.92 2.93 1.70587 1.70880 1.71172 5.39444 5.40370 5.41295 24.6422 24.8971 25.1538 1.42768 1.42931 1.43094 3.07584 3.07936 3.08287 6.62671 6.63429 6.64185 8.4681 8.5261 8.5849 8.6436 8.7025 8.7616 2.94 2.95 2.96 1.71464 1.71756 1.72047 5.42218 5.43139 5,44059 1.43257 1.43419 1.43581 3.08638 3.08987 3.09336 25.4122 25.6724 25.9343 26.1981 20.4636 26.7309 6.64940 6.65693 6.66444 6.671.94 6.67942 6.68688 2.97 2.98 2.99 8.8209 8.8804 8.9401 1.72337 1.72627 1.72916 5.44977 5.45894 5.46809 1.43743 1.43204 1.44065 3.09684 3.10031 3.10378 - ·il) Powers and Roots xiii n vn ✓1012 in V10 n $100 n 8.00 9.000 1.73205 5.-17723 27.0000 1.44220 3.10723 6.69433 3.01 3.02 3.03 9.0601 9.120+ 9.1809 1.73491 1.73781 1.74069 5.486:30 5.19545 5.50-54 27.2709 27.5430 27.8181 1.1.1385 1.41515 1.44704 3.11068 3.11412 3.11756 6.70176 6.70917 6.71677 3.04 3.05 3.00 9.2016 9.3025 9.3636 1.74356 1.74642 1.74929 5.51302 5.52268 5.53173 28.0940 28.3726 28.6526 1.41863 1.45022 1.45180 3.12098 3.12.10 3.12781 6.72390 6.73132 6.73866 6.74000 6.75331 6.76061 3.07 3.08 3.09 9.4249 9.4864 9.5481 1.75214 1.75499 1.75784 5.54076 5.54977 5.558T8 28.9344 29.2181 29.5036 1.15338 1.45-196 1.45033 3.13121 3.13401 3.1:3800 3.10 9.6100 1.70068 5.50770 29.7910 1.45810 3.14138 6.76790 3.11 3.12 3.13 9.6721 9.7344 9.7969 1.70352 1.70035 1.70918 5.57674 5.58570 5.59464 30.0802 30.3713 30.6643 1.45:07 1.40123 1.40279 3.14475 3.14812 3.15148 6.77517 6.78242 6.78966 3.11 3.15 3.16 9.8596 9.9225 9.9856 1.77200 1.77482 1.7776+ 5.60357 5.61219 5.62139 30.9591 31.2559 31.5545 1.40434 1.405.90 1.40745 3.15483 3.15818 3.16152 6.79688 6.80109 6.81128 3.17 3.18 3.19 10.0189 10.1121 10.1761 1.78045 1.78326 1.78606 5.63028 5.63915 5.64801 31.8550 32.1574 32.4018 1.468X) 1.47051 1.47208 3.16485 3.16817 3,17149 6.81846 6.82562 6.83277 3.20 10.2400 1.78885 32.7080 1.47361 3.17480) 6.83990 3.21 3.22 3.23 10.3041 10.3081 10.4329 1.79165 1.79141 1.79722 5.65085 5.665569 5.67450 5.68331 33.0762 33.3802 33.6183 1.47515 1.47008 1.47820 3.17811 3.18110 3.18169 6.84702 6.80112 6.86121 3.24 3.25 3.26 10.4976 10.5025 10.0276 1.80000 1.80278 1.80555 5.69210 5.70088 5.70964 34.0192 34.3281 31.6160 1.47973 1.48125 1.48277 3.187.8 3.19125 3.1952 6.86829 6.87534 6.88239 3.27 3.28 3.29 10.6929 10.7581 10.8241 1.80831 1.81108 1.81384 5.71839 5.72713 5.73585 34.2058 35.2876 35.6113 1.48428 1.48579 1.48780 3.19778 3.20104 3.20429 6.88942 6.8443 6.90344 3.30 10.8%) 1.81659 5.744456 35.9370 1.48881 3.20703 0.91042 3.31 3.32 3.33 10..9561 11.022+ 11.0889 1.81934 1.84209 1.82483 5.75:326 5.7619+ 5.77062 36.2017 36.5:41 36.9260 1.4831 1.49181 1.49:330 3.21077 3.21400 3.21722 6.91740 6.92436 6.93130 3.31 3.3 3.36 11.1556 11.2:25 11.28.16 1.82757 1.83030 1.83303 5.77927 5.78792 5.79655 37.2597 37.5954 37.9331 1.49480 1.4920 1.49777 3.22044 3.22305 3.22086 6.93823 6.94515 6.95205 3.37 3.38 3.39 3.40 11.3569 11.7214 11.1921 1.83576 1.83818 1.84120 5.80517 5.81378 5.82237 38.2728 38.6145 38.9982 39.3040 1.49926 1.50074 1.50222 3.23006 3.23323 3.23643 6.95894 6.96582 6.97268 11.5000 1.84391 5.83095 1.50309 3.23901 6.97953 3.41 3.42 3.43 1.81602 1.81932 1.85203 5.82952 5.81808 5.85662 39.0518 40.0017 40.3536 1.50517 1.50604 1.50810 3.24278 3.24595 3.24911 6.98637 6.21319 7.00000 11.6281 11.694 11.7649 11.8336 11.9025 11.9716 3.44 3.45 3.46 1.85172 1.85742 1.86011 40.7076 41.00:36 41.4217 1.50957 1.51103 1.51249 3.25227 3.25542 3.25856 7.00680 7.01358 7.02035 5.86515 5.87367 5.88218 8.891067 5.89915 5.702 3.47 3.48 3.49 12.0409 12.1104 12.1801 1.86279 1.86048 1.86813 41.7819 42.1442 42.5085 1.51394 1.51540 1.51685 3.26169 3.26482 3.26795 7.02711 7.03385 7.04058 ·XIV Powers and Roots 111 n vi V10 n ns V10 n 100 n 8.50 12.2700 1.87083 5.91608 42.8750 1.51829 3.71 3.52 3.53 12.3201 12.3204 12.4609 1.87350 1.87017 1.87883 5.92433 5.9326 5.94138 43,24:30 43.6142 43.9870 1.51974 1.52118 1.52262 3.27107 3.27418 3.27729 3.28039 7.04730 7.05400 7.06070 7.06738 3.54 3.55 3.56 12.5316 12.6025 12.6736 1.88149 1.88414 1.88680 5.94979 5.95819 5.96607 1.52406 1.52549 1.52692 3.28348 3.28057 3.28965 7.07404 7.08070 7.08734 44.3019 41.7389 45.1180 45.4993 45.8827 46.2683 3.57 3.58 3.59 12.7479 12.8161 12.8881 12.9600 1.88944 1.89209 1.89173 5.97495 5.98331 5.99166 1.52835 1.52978 1.53120 7.09397 7.10059 7.10719 7.11379 1.89737 6.00000 46.6500 1.53202 3.60 3.01 3.02 3.63 13.0321 13.101 13.1769 1.0000 1.90203 1.90526 47.0459 47.4379 47.8321 1.53404 1.53545 1.53686 0.00833 6.0166+ 6.02495 6.03324 0.01152 6.04979 3.29273 3.29580 3.20887 3.30193 3.30198 3.30803 3.31107 3.31411 3.31714 3.32017 7.12037 7.12694 7.13349 7.14004 7.14057 7.10309 3.04 3.65 3.66 13.2496 13.3225 13.3956 1.90788 1.91050 1.91311 48.2285 48.6271 49.0279 1.53827 1.53X68 1.04109 3.67 3.68 3.69 13.4689 13.5124 13.6161 1.91572 1.91833 1.92091 1.92357 6.05805 6.00030 0.07451 49.4309 49.8300 50.2434 50.05.30 1.54249 1.54389 1.51529 3.32319 3.32021 3.32922 7.1590 7.16610 7.17258 13.6900 6.08270 1.04068 3.33222 7.17 9001 3.70 3.71 3.72 3.73 13.7041 13.8384 13.9129 6.00098 6.09018 6.10737 51.0618 51.1788 51.8951 1.54807 1.51910 1.55085 3.74 3.75 3.76 1.92614 1.92873 1.93132 1.93391 1.93019 1.93907 1.91165 1.94122 1.91679 1.94936 7.18552 : 7.191971 7.19840 7.20483 7.21125 7.21765 13.9876 14.0025 14.1376 6.11553 6.12372 6.13188 6.14003 6.14817 6.15030 52.3136 02.7347 53.1574 53.5826 54.0102 54.4399 3.33322 3.33822 3.34120 3.34419 3.31716 3.35014 3.35310 3.35007 3.35202 1.55223 1.53302 1.55500 1.55037 1.55775 1.55912 3.77 3.78 3.79 14.2129 14.2884 14.3011 7.22405 : 7.23043 7.23680 3.80 14.4400 6.16111 54.8720 1.56049 3.36198 7.2-43101 3.81 3.82 3.83 14.5101 14.5924 14.6689 1.95192 1.95418 1.95701 6.172752 0.18001 6,18870 55.3003 55.7430 56.1819 1.56186 1.56322 1.56459 7.24930 7.25581 7.20217 3.36192 3.30786 3.37080 3.37373 3.37666 3.37958 1.95959 1.X214 1.96169 6.19677 6.20484 6.21289 50.6231 57.0606 57.5125 1.50595 1.50731 1.56866 3.81 3.85 3.86 3.87 3.88 3.89 14.7450 14.8225 14.8996 14.9769 15.0544 15.1321 7.26848 7.27479 7.28108 7.28736 7.29303 7.29989 1.96723 1.8977 1.97231 6.22093 6.22896 6.23699 57.2006 58.1111 58.8639 1.07001 1.57137 1.57271 3.38249 3.38540 3.38831 3.90 15.2100 1.97484 6.24500 59.31.90 1.57406 3.39121 7.30614 15.2881 15.3664 15.4419 6.25300 6.20099 6.26897 59.7765 (0.2363 60.0085 1.57511 1.57675 1.57809 3.39411 3.39700 3.39988 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 1.97737 1.97990 1.98242 1.98494 1.98746 1.98997 7.31238 7.31861 7.32483 7.33104 7.33723 7.34342 15.5236 15.6025 15.6816 61.1030 61.0299 62.0.991 1.87942 1.58076 1.58209 6.27644 6.28490 6.29285 6.30079 6.30872 6.31664 3.40277 3.40564 3.40851 3.41138 3.41424 3.41710 18.7609 15.8404 15.9201 1.99249 1.99498) 1.99750 62.5708 63.0448 63.5212 1.58342 1.58475 1.58608 7.34960 7.35576 7.36192 ·II] Powers and Roots XV n vn V10 n n8 in $10 n 100 na 4.00 16.0000 2.00000 6.32456 04.0000 1.58740 3.41995 4.01 4.02 4.03 16.0801 16.1604 16.2409 2.00200 2.00499 2.00749 1.58872 1.5.2004 1.59136 3.42280 3.42504 3.42848 7.36806 7.37420 7.380:2 7.38644 6.332-16 6.310:35 6.34823 6.35610 6.30396 6.37181 64.4812 64.9648 65.4508 65.9393 66.4301 60.9234 4.04 7.05 4.06 16.3216 10.4025 16.4836 2.00998 2.01246 2.01494 1.59267 1.59399 1.59530 3.43131 3.43414 3.43697 7.39254 7.39864 7.40472 7.41080 7.41686 7.42291 4.07 4.08 4.09 16.5649 10.6464 10.7281 2.01742 2.01990 2.02237 6.37966 6.38749 6.39531 67.4191 67.9173 68.4179 1.59661 1.59791 1.59922 3.43979 3.44260 3.415411 4.10 16.8100 2.02485 6.40312 68.9210 1.60052 3.44822 7.4282 4.11 4.12 4.13 16.8921 16.9744 17.0569 2.02731 2.02978 2.03224 6.41093 6.41872 6.42651 09.4265 09.0345 70.4450 1.60182 1.60312 1.60411 7.43499 7.44102 7.44703 3.45102 3.45382 3.45661 3.45939 3.46218 3.40496 4.14 4.15 4.16 17.1326 17.2225 17.3056 2.03470 2.03715 2.03961 6.43128 6.44205 6.44981 70.9579 71.4734 71.9913 1.60571 1.60700 1.60829 7.45.704 7.450004 7.46502 4.17 4.18 4.19 4.20 17.3889 17.4724 17.5561 2.07206 2.04150 2.04695 6.45755 6.46529 6.47302 72.5117 73.0346 73.5601 1.60956 1.61086 1.61215 3.46773 3.47050 3.47327 7.47100 7.47697 7.48292 7.48887 17.6100 2.04939 6.48074 74.0880 1.01343 3.47603 4.21 4.22 4.23 17.7241 17.8084 17.8929 2.05183 2.05126 2.05070 6.48845 6.4.2015 6.50384 74.6185 75.1514 75.6870 1.61471 1.61599 1.61726 3.47878 3.4815- 3.48128 7.4.9481 7.50074 7.50666 17.9776 18.0625 18.1476 2.05913 2.06155 2.00398 6.51153 6.51920) 6.52087 76.2250 76.7650 77.3088 1.61853 1.619981 1.02108 3.48703 3.48977 3.492550 4.24 4.25 4.26 4.27 4.28 4.29 7.51257 7.51847 7.52437 7.53025 7.53612 7.5-1199 18.2329 18.3184 18.1011 2.06610 2.06882 2.07123 6.53452 6.54217 6.54981 77.8545 78.4028 78.9536 1.62234 1.62361 1.62487 3.49123 3.49796 3.50008 4.30 18.400 2.07361 6.55744 79.1070 1.02013 3.50340 7.5-1784 4.31 4.32 4.33 18.5761 18.6621 18.7489 6.56700 6.57207 6.58027 80.0630 80.6216 81.1827 3.50611 3.50882 3.51153 2.07605 2.07846 2.08087 2.08327 2.08567 2.08806 1.62739 1.02865 1.62991 1.63116 1.63241 1.63366 4.34 4.35 4.36 18.8356 18.9225 19.0046 6.58787 6.59545 6.60303 81.7465 82.3129 82.8819 3.51423 3.51092 3.51062 7.55369 7.55.953 7.50535 7.57117 7.57698 7.58279 7.58858 7.5.9436 7.60014 4.37 4.38 4.39 4.40 19.0969 19.1814 19.2721 2.09045 2.09284 2.09323 6.61060 6.61816 6.02571 83.4535 84.0277 84.6045 1.63491 1.03619 1.03740 3.92231 3.52499 3.52767 19.3600 2.09762 6.63325 85.1840 3.53035 7.60590 4.41 4.42 4.43 19.4181 19.5364 19.6249 2.10000 2.10238 2.10476 6.64078 6.64831 6.65582 85.7661 86.3709 80.9383 1.63864 1.03988 1.04112 1.64236 3.53302 3.53509 3.53835 7.61166 7.61741 7.62315 4.44 4.45 4.46 7.62888 7.63461 7.61032 19.7136 19.8025 19.8916 19.9809 20.0704 20.1601 2.10713 2.10950 2.11187 2.11424 2.11660 2.11896 6.66333 6.67083 6.67832 6.68581 6.69328 6.70075 87.5284 88.1211 88.7165 89.3146 89.9154 90.5188 1.64359 1.64483 1.64606 1.64729 1.64851 1.64974 3.54101 3.51307 3.54632 3.84897 3.55102 3.55426 4.47 4.48 4.49 7.64603 7.05172 7.65741 ·XV1 Powers and Roots (1) n na vn V10 n nes 4.50 20.2500 2.12132 6.70820 91.1250 1.65096 4.51 4.52 4.53 20.3401 20.4304 20.5209 2.12368 2.12603 2.12838 6.71505 6.72309 6.73053 91.7339 92.34 92.9597 1.0.7219 1.05311 1.65402 4.84 4.55 4.56 20.6116 20.7025 20.7936 2.13073 2.13307 2.13542 93.5767 94.1964 94.8188 1.65584 1.65706 1.65827 6.73795 6.74537 6.75278 6.76018 6.76757 6.77495 4.57 4.58 4.59 2.13776 2.14009 2.14243 1.65148 1.06069 1.66190 6.78233 1.00310 10 n 1100 n 3.50689 7.66309 3.55953 7.66877 3.50215 7.67443 3.50478 7.68009 3.56740 7.68573 3.57002 7.69137 : 3.57263 7.09700 3.57524 7.70202 3.57785 7.70824 3.58045 7.71384 3.58305 7.71944 3.58504 7.72503 3.78823 7.73061 3.59082 7.73619 3.59340 7.74175 3.39598 7.74731 3.59856 7.75286 20.8849 20.9704 21.0681 21.1000 21.2721 21.314 21.4369 21.52:6 21.0225 21.7156 4.60 4.61 4.02 4.63 2.14476 2.14709 2.14912 2.15174 95.4440 96.0719 96.7026 97.3300 97.9722 98.6111 99.2528 99.8973 100.515 101.105 6.78970 6.79700 6.80441 1.66431 1.66551 1.66671 1.66791 1.00911 1.67030 4.64 4.05 4.06 2.15-107 2.15039 2.15870 6.81175 6.81.909 6.82012 4.67 4.08 4.09 21.8089 21.9024 21.921 2.16102 2.10333 2.10564 0.83374 6.81105 6.84836 101.848 102.503 103.102 1.67150 1.67209 1.07388 3.60113 3.60370 3.60026 7.75840 7.76394 7.70440 4.70 02.090 2.10795 6.87565 103.823 1.67507 3.60883 7.771998 4.71 4.72 4.73 22.1841 22.2784 22.3729 2.17025 2.17250 2.17-186 6.80294 6.87023 6.87750 104.487 105.154 105.824 1.07020 1.67744 1.67803 3.61138 3.01324 3.61619 7.78049 7.78599 7.79149 4.74 4.75 4.76 22.4676 22.5625 22.6576 2.17715 2.17915 2.18174 6.88477 6.891202 6.89928 1.07981 1.08099 1.68217 3.61903 3.02158 3.02112 7.70697 7.80245 7.80793 4.77 4.78 4.79 22.7529 22.8184 22.2441 2.18403 2.180:12 2.18861 6.90652 6.91375 6.92098 6.92820 106.476 107.172 107.850 108.531 109.215 100.202 110.592 1.68334 1.68452 1.68509 1.08087 3.62065 3.62919 3.03172 4.80 23.0400 2.1.9089 3.63124 4.81 4.82 4.83 23.1301 23.2324 23.3289 2.19317 2.19515 2.19773 6.93542 6.94202 6.94982 111.285 111.980 112.079 1.68804 1.08920 1.09037 3.63676 3.63928 3.61180 3.64431 3.64682 3.64932 7.81339 7.81885 7.82129 7.82974 7.83517 7.81059 7.81601 7.85142 7.85683 7.86222 4.84 4.85 4.86 23.4256 23.5225 23.6196 2.20000 2.20227 2.20454 6.95701 6.26419 6.97137 113.380 114.081 114.791 1.69154 1.69270 1.69386 4.87 4.88 4.89 23.7169 23.8147 23.9121 2.20681 2.2017 2.211:33 6.97854 6.98070 6.99285 115.501 116.214 116.930 1.69503 1.6.2019 1.69734 3.60182 3.65432 3.65681 7.86761 7.87299 7.87837 4.90 24.0100 2.21359 7.00000 117.649 1.69850 3.65931 7.88374 4.91 4.92 4.93 24.1081 24.2064 24.3019 2.21585 2.21811 2.22036 118.371 119.0995 119.823 1.099135 1.70081 1.70196 7.889309 7.89445 7.89979 4.94 4.95 4.96 2.22261 2.22486 2.22711 7.00714 7.01427 7.02140 7.02851 7.03562 7.04273 7.04982 7.05691 7.06399 120.0154 121.287 122.024 24.4036 24.5025 24.6016 24.7009 24.8004 24.9001 7.90513 7.91046 7.91578 3.66179 3.66428 3.66676 3.66924 3.67171 3.67418 3.67665 3.07911 3.68157 1.70311 1.70426 1.70540 1.70655 1.70769 1.70884 4.97 4.98 4.99 2.22935 2.23159 2.23383 122.763 123.50% 124.251 7.92110 7.92641 7.93171 ·II] Powers and Roots xvii n Vri ✓10 n yn $10 n 100 n 5.00 2.23007 7.07107 125.000 1.70998 3.68103 7.93701 5.01 5.02 5.03 25.0000 253.1001 25.2004 25.3009 2.2:3830 2.240.34 2.24277 7.07814 7.08020 7.09225 125.752 120.506 127.204 1.71112 1.71225 1.71339 3.68679 3.08894 3.69138 7.94229 7.94757 7.95285 5.04 5.05 0.06 25.4016 25.5025 25.6036 7.09930 7.10634 7.11337 128.024 128.788 129.554 3.69383 3.6.7627 3.09871 7.95811 7.3337 7.96863 2.24199 2.21722 2.24944 2.25107 2.25389 2.25610 1.71452 1.71566 1.71679 1.71792 1.71909 1.72017 5.07 5.08 5.09 25.7049 25.8061 25.8081 7.12039 7.12741 7.13442 130.324 131.097 131.872 3.70114 3.70357 3.70600 7.97387 7.97911 7.984:34 5.10 3.70813 7.98957 5.11 5.12 5.13 26.0100 26.1121 26.2144 26.3169 2.25832 2.26053 2.20274 2.26495 7.14143 7.14813 7.16.542 7.102-10 7.10938 7.170) 7.18331 132.651 133,133 131.218 135.000 1.72130 1.72242 1.72355 1.72467 3.71085 3.71327 3.71569 3.71810 3.72051 3.72292 7.99479 8.000000 8.00520 8.01010 8.01589 8.02078 5.14 5.15 5.16 20.1126 20.5225 26.6256 2.26716 2.28936 2.27156 135.797 130.591 137.388 1.72579 1.72091 1.72802 5.17 5.18 5.19 5.20 26.7289 26.8324 26.9361 2.27376 2.273986 2.27816 7.1.9027 7.19722 7.20417 138.188 1:38.992 139.798 1.72914 1.73023 1.73137 1.73248 3.72532 372772 3.73012 8.02526 8.03113 8.03629 27.0400 2.28030 7.21110 1410.608 3.733251 8.04145 5.21 5.22 5.23 27.1411 27.2484 27.3529 2.28274 2.28173 2.28692 7.21803 7.224.46 7.23187 141.421 142.237 1.13.050 1.73359 1.73170 1.73580 3.73130 3.73729 3.73768 8.04600 8.05175 8.07689 5.24 5.25 5.26 143.878 144.703 145.532 1.7:3691 1 7:3801 1.73912 8.00202 8.06714 8.07226 27.4576 27.5625 27.0676 27.7729 27.8784 27.9841 2.28910 2.2.129 2.29347 2.29565 2.29783 2.30000 7.23878 7.24509 7.27209 7.25948 7.20036 7.27324 3.74206 3.7444: 3.74681 3.74918 :3.75155 3.75392 5.27 5.28 5.29 146.363 117.198 148.036 1.74022 1.74132 1.74242 8.07737 8.08248 8.08758 5.30 28.0XX) 2.30217 7.28011 148.877 1.74351 8.09267 5.31 5.32 5.33 28.1961 28.3024 28.4089 2.30431 2.30651 2.30868 7.28097 7.29:383 7.30068 119.721 150.509 151.419 1.74461 1.74570 1.74680 3.75629 3.75865 3.76101 3.70336 8.09776 8.10284 8.10791 5.34 5.35 5.36 28.5156 28.6225 28.722 2.31081 2.31301 2.31517 7.30753 7.31437 7.32120 152.273 153.1:30 153.991 1.74789 1.74898 1.75007 3.76771 3.76806 3.77041 8.11298 8.11804 8.12310 5.37 5.38 5.39 5.40 28.8369 28.9444 29.0521 2.31733 2.31918 2.32104 7.32803 7.33485 7.34166 154.854 155.721 156.591 1.75116 1.75221 1.75333 3.77275 3.77509 3.77743 8.12814 8.13319 8.13822 29.1000 2.32379 7.34847 1.75441 3.77976 8.14325 1.41 5.42 5.43 7.35527 7.36206 7.36885 167.461 158.340 15.9.220 160.103 1.75549 1.75057 1.75765 3.78209 3.78442 3.78675 8.14828 8.15329 8.16831 29.2681 29.3704 29.4849 29.5936 29.7025 29.8116 2.32594 2.32809 2.33024 2.33238 2.33452 2.83666 5.44 5.45 5.46 8.16331 8.16831 8.17330 7.37504 7.38241 7.38918 7.39594 7.40270 7.40945 160.989 161.879 162.771 163.667 104.567 105.469 1.75873 1.75981 1.76088 1.76196 1.76303 1.76110 3.78207 3.79139 3.79371 3.79603 3.79834 3.80065 5.47 5.48 5.49 29.9209 30.0304 30.1401 2.33880 2.34094 2.34307 8.17829 8.18327 8.18824 ·xviii Powers and Roots n n2 vñ V10 n ns V10 n +100 n 30.2500 7.41620 166.375 1.76517 3.80295 8.19321 5.50 5.51 5.52 5.53 2.34521 2.34734 2.34947 2.35160 2.35372 2.35584 2.35797 167.284 168.197 169.112 1.76624 1.76731 1.76838 30.3601 30.4704 30.5809 30.6916 30.8025 30.9136 31.0249 31.1364 31.2481 5.54 5.55 5.56 7.42294 7.42967 7.43640 7.44312 7.44983 7.45054 7.46324 7.463M 7.47663 1.7694+ 1,77051 1.77157 170.031 170.954 171.880 172.809 173.741 174.677 3.80526 8.19818 3.80756 8.20313 3.80985 8.20808 3.81215 8.21303 3.81444 8.21797 3.81673 8.2229) 3.81902 8.22783 3.82130.8.23275 3.82358 8.23766 3.82786 8.24257 5.57 5.58 5.59 2.36008 2.36220 2.36732 1.77263 1 77369 1.77475 5.60 31.3600 7.48331 1.77581 5.61 5.62 5.63 31.4721 31.5844 31.6969 2.36613 2.36854 2.37065 2.37276 7.48999 7.4807 7.50333 175.616 176.558 177.501 178.454 179.406 180.362 181.321 1.77686 1.77792 1.77897 1.78003 1.78108 1.78213 3.82814 3.83041 3.83268 3.83495 3.83722 3.83948 8.24747 8.25237 8.25726 8.26215 8.20703 8.27190 5.64 5.65 5.66 31.8096 31.9225 32.0356 2.37487 2.37697 2.37908 7.50999 7.51605 7.52330 5.67 5.68 5.69 32.1489 32.2624 32.3761 2.38118 2.38328 2.38537 7.52994 7.53658 7.54321 182.284 183.250 184.220 185.193 1.78318 1.78422 1.78527 3.84174 3.84399 3.81625 8.27677 8.28164 8.28649 5.70 32.4.200 2.38747 7.51983 1.78632 3.84850 8.2:)134 5.71 5.72 5.73 32.6041 32.7184 32.8329 186.169 187.149 188.133 1.78736 1.78840 1.78944 3.85075 3.85300 3.85524 8.2:619 8.30103 8.30587 2.35956 2.39165 2.39374 2.39583 2.39792 2.40000 7.556 15 7.50:307 7.56968 7.57028 7.58288 7.58917 5.74 5.75 5.76 32.9476 33.0025 33.1776 1.79048 1.79152 1.79256 3.80748 3.85972 3.86190 8.31069 : 8.31552! 8.32034 189.119 100.109 191.103 192.100 193.101 194.105 5.77 5.78 5.79 33.2929 33.4084 33.5241 2.40208 2.40416 2.40624 7.59605 7.00203 7.00920 1.79360 1.70163 1.79567 1.7.2670 3.86119 3.86642 3.86865 8.32515 8.32995 8.33476 8.33955 6.80 2.40832 195.112 3.87088 5.81 5.82 6.83 33.0400 33.7501 33.8724 33.9889 2.41039 2.41247 2.41454 7.61577 7.62234 7.62889 7.63544 7.64199 7.64853 7.65506 196.123 197.137 198.155 1.79773 1.79876 1.79979 3.87310 3.87532 3.87754 8.34434 8.34913 8.35390 5.84 5.85 5.86 34.1056 34.2225 34.3396 2.41661 2.41868 2.42074 199.177 200,202 201.230 1.80082 1.80185 1.80288 3.87975 3.88197 3.88418 8.358681 8.36345 8.36821 5.87 5.88 0.89 34.4569 34.5744 34.6921 34.8100 2.42281 2.42487 2.42693 7.66159 7.66812 7.67463 7.68115 202.262 203,297 204.3:36 1.80390 1.80492 1.80595 3.88639 3.88859 3.8.2080 8.37297 8.37772 8.38247 5.90 2.42899 205.379 1.80697 3.89300 8.38721 5.91 5.92 5.93 34.9281 35.0464 35.1649 2.43105 2.43311 2.43516 206.425 207.475 208.028 1.80799 1.80001 1.81003 3.89519 3.89739 3.89958 8.39194 8.36667 8.40140 209.585 210.645 211.709 1.81104 1.81206 1.81307 7.68765 7.6415 7.70065 7.70714 7.71362 7.72010 7.72658 7.73305 7.73:151 3.90177 3.90396 3.90618 5.94 5.96 8.96 8.97 5.98 5.99 8.40612 8.41083 8.41564 35.2836 35.4025 35.5216 35.6409 35.7604 35.8801 2.43721 2.49926 2.44131 2.44336 2.44540 2.44745 212.776 213.847 214.922 1.81409 1.81510 1.81611 3.90833 3.91051 3.91269 8.42025 8.42494 8.42964 ·II] Powers and Roots xix n2 vn ✓10 n n3 310 n. $100 n 6.00 36.0000 2.44949 7.74597 1.81712 6.01 6.02 6.03 36.1201 36.2404 36.3609 2.45153 2.45357 2.45561 7.75242 7.75887 7.76531 216.000 217.082 218.107 219.256 1.81813 1.81914 1.82014 8.43433 8.43901 8.44369 8.44836 6.04 6.05 6.06 36.4816 36.6025 36.7236 2.45764 2.45967 2.46171 7.77174 7.77817 7.78460 220.349 221.445 222.545 1.82115 1.80215 1.82316 3.91187 3.91704 3.91921 3.92138 3.92355 3.92571 3.92787 3.93003 3.93219 3.93434 8.45303 8.45769 8.46235 6.07 6.08 6.09 36.8449 36.4664 37.0881 2.46374 2.40577 2.46779 7.7.102 7.79714 7.80385 223.049 224.750 225.807 1.82416 1.82510 1.82610 8.40700 8.47165 8.47029 8.48093 37.2100 7.81025 226.981 1.82716 3.93050 6.10 6.11 6.12 6.13 37.3321 37.4544 37.5769 2.46982 2.47184 2.47386 2.47588 7.81665 7.82304 7.82943 228.099 229.221 230.316 3.93805 3.44079 3.44294 1.82816 1.82915 1.83015 1.83115 1.83214 1.83313 8.48550 8.4.2018 8.19481 8.49442 8.50403 8.50864 6.14 6.15 6.16 37.6996 37.8225 37.9456 2.47790 2.47992 2.48193 231.476 232.008 233.745 7.83582 7.84219 7.84857 7.81193 7.86130 7.86766 3.94508 3.994722 3.94936 3.95150 3.95303 3.95576 6.17 6.18 6.19 38.0689 38.1924 38.3161 2.48395 2.48536 2.48797 234.885 230.029 237.177 1.83412 1.83511 1.83610 8.51324 8.51784 8.52243 6.20 38.4400 2.48998 7.87401 238.328 1.83709 3.95789 8.52702 6.21 6.22 6.23 38.5641 38.6884 38.8129 2.49199 2.49399 2.49600 7.88036 7.88670 7.89303 2:39.483 240.042 241.804 1.83808 1.8:39106 1.81005 3.2002 3.3214 3.96427 8.53100 8.533618 8.51075 6.24 6.25 6.26 38.9376 39.0625 39.1876 2.49800 2.50000 2.50200 7.899:37 7.X509 7.91202 242.971 241.141 245.314 1.84103 1.84202 1.84300 8.51532 8.01988 8.55444 3.96638 3.26850 3.97062 3.97273 3.97-184 3.97095 6.27 6.28 6.29 39.3129 39.4384 39.5041 2.50400 2.505.99 2.50799 7.91833 7.92405 7.93095 246.492 247.073 248.858 1.84398 1.84436 1.8.1991 8:55899 8.56354 8.56808 6.30 39.6.200 2.50998 7.9:3725 250.047 1.81691 3.97906 8.57262 6.31 6.32 6.33 39.8161 39.9424 40.0689 2.51197 2.51356 2.51595 7.94355 7.94984 7.95613 251.240 252.436 253.636 1.84789 1.84887 1.84984 3.98116 3.98320 3.98536 8.57715 8.58168 8.58620 6.34 6.35 6.36 40.1956 40.32:25 40.44% 2.51794 2.519.2 2.5219 7.96241 7.96869 7.97496 254.840 256.018 257.259 1.85082 1.85179 1.85276 3.98746 3.98956 3.99165 8.59072 8.5.9524 8.59975 6.37 6.38 6.39 40.5769 40.7044 40.8321 2.52389 2.52787 2.52784 7.98123 7.98749 7.99375 258.475 259.0014 260.917 1.85373 1.85470) 1.87507 3.99374 3.99583 3.93702 8.60425 8.60875 8.61325 6.40 40.9600 2.52982 8.0000) 262.144 1.85064 4.00000 8.61774 6.41 6.42 6.43 41.0881 41.2164 41.3449 2.53180 2.53377 2.533574 8.00625 8.01249 8.01873 203.375 261.609 265.848 1.85760 1.85857 1.85953 4.00208 4.00410 4.00624 8.62222 8.62671 8.63118 6.44 6.45 6.46 41.4736 41.6025 41.7316 2.53772 2.539469 2.54165 8.02496 8.03119 8.03741 267.000 268.336 269.586 4.00832 4.01039 4.01246 8.63566 8.64012 8.64459 1.86050 1.86146 1.86242 1.80338 1.86434 1.86530 6.47 6.48 41.8009 41.94X4 12.1201 2.54362 2.54558 2.54755 8.04303 8.04984 8.05605 270.840 272.098 273.359 4.01453 4.01660 4.01866 8.04904 8.65350 8.65795 6.49 ·vn 1.88801 / 4 XX Powers and Roots [II n n2 ✓10 n ns 310 n *100 n 6.50 42.2500 2.54951 8.06226 274.025 1.86026 4.02073 8.60239 6.51 42.3801 2.55147 8.06846 275.894 1.86721 4.02279 8.06683 6.52 42.5104 2.55343 8.07465 277.168 1.86817 4.02485 8.07127 6.53 42.6409 2.55539 8.08084 278.445 1.86912 4.02690 8.67570 6.54 42.7716 2.55734 8.08703 279.726 1.87008 4.02896 8.68012 6.55 42.2025 2.55930 8.09321 281.011 1.87103 4.03101 8.68455 6.56 43.0336 2.56125 8.09938 282.300 1.87198 4.03306 8.68896 6.57 43.1649 2.56.320 8.10555 283.593 1.87293 4.03511 8.69338 6.58 13.2964 2.56515 8.11172 284.8 1.87388 4.03715 8.69778 6.59 43.4281 2.56710 8.11788 280.191 1.87-483 4.03920 8.70219 6.60 43.5600 2.50.105 8.12404 287.496 1.87578 4.041244 8.70659 6.61 43.6921 2.57099 8.13019 288.805 1.87072 4.04328 8.71098 6.62 43.8244 2.57291 8.13634 20.118 1.87767 4.04532 8.71537 6.63 43.9569 2.57.188 8.14248 291.434 1.87862 4,04735 8.71976 6.64 44.0896 2.57682 8.14862 292.755 1.87956 4.04939 8.72414 6.65 44.2225 2.57876 8.15475 294.080 1.88050 4.05142 8.72852 6.66 44.3556 2.58070 8.16088 295.408 1.88144 4.05345 8.73289 6.67 44.4889 2.58263 8.16701 296.741 1.88239 4.051518 8.73726 6.68 41.6224 2.58457 8.17313 298.078 1.88333 4.05750 8.74162 6.69 44.7561 2.58650 8.17924 290.418 1.88127 4.07953 8.74598 6.70 47.80%O 2.58844 8.18.535 300.763 1.88520 4.06155 8.75034 6.71 45.0241 2.58037 8.19146 302.112 1.88614 4.00:357 8.75409 6.72 45.1584 2.50230 8.19756 303.164 1.88708 4.06759 8.75904 6.73 45.2929 2.59422 8.20366 304.821 4.06760 8.76338 6.74 45.4276 2.59615 8.20975 306.182 1.88895 4.06961 8.76772 6.75 45.5025 2.5.9808 8.21584 307.547 1.88988.1 4.07103 8.77205 6.76 45.6976 2.00000 8.22192 308.916 1.80081 4.07364 8.77638 6.77 45.8329 2.60192 8.22800 310.289 1.89175 4.07064 8.78071 6.78 45.681 2.00384 8.23108 311.666 1.89268 4.07765 8.78503 6.79 46.10+1 2.60576 8.24015 313.047 1.80361 4.07965 8.78935 6.80 46.2400 2.60768 8.24621 314.432 1.89454 4.08166 8.79366 6.81 46.3761 2.60960 8.25227 315.821 1.89546 4.08365 8.79797 6.82 46.5124 2.61151 8.25833 317.215 1.8)39 4.08565 8.80227 6.83 46.6489 2.61313 8.20438 318.612 1.89732 4.08765 8.80657 6.84 46.7856 2.61534 8.27043 320.014 1.89824 4.08964 8.81087 6.85 46.9225 2.61725 8.27647 321.419 1.89917 4.09163 8.81516 6.86 47.05.16 2.61916 8,28251 322.829 1.90009 4.09362 8.81945 6.87 47.1969 2.62107 8.28855 324.243 1.90102 4.09561 8.82373 6.88 47.3344 2.62298 8.23458 325.061 1.90194 4.09700 8.82801 6.89 47.4721 2.62188 8.30000 327.083 1.90286 4.09958 8.83228 8.90 47.6100 2.62679 8.30662 328.509 1.90378 4.10157 8.83656 6.91 47.7481 2.02869 8.31204 329.939 1.90470 4.10355 8.84082 6.92 47.8864 2.63059 8.31865 3:31.374 1.90562 4.10552 8.84509 6.93 48.0249 2.63249 8.32466 332.813 1.00053 4.10750 8.84934 6.94 48.1636 2.63439 8.33007 334.255 1.90745 4.10948 8.85360 6.955 48.3025 2.63629 8.33667 335.702 1.90837 4.11145 8.85785 6.96 48.4416 2.63818 8.34266 337.154 1.90928 4.11342 8.86210 6.97 48.5809 2.64008 8.34865 338.609 1.91019 4.11539 8.86634 : 6.98 48.7204 2.64197 8.35464 310.068 1.91111 4.11736 8.870581 6.99 48.8601 2.64386 8.36062 341.532 1.91202 4.11932 8.87481 ·II] Powers and Roots xxi n na vn ✓10 n in 10 n 100 n. 7.00 49.0000 8.36660 343.000 1.91293 4.12129 8.87904 7.01 7.02 7.03 19.1401 49.2804 49.4209 2.61575 2.64704 2.61053 2.05141 344.472 315.948 347.429 4.12325 4.12521 4.12716 8.88327 8.88749 8.89171 8.37257 8.37854 8.38151 8.39047 8.39013 8.40238 1.91384 1.91475 1.91566 1.91657 1.91747 1.91838 49.5616 49.7025 49.8436 2.65330 2.05518 2.65707 348.914 350.403 351.86 4.12912 4.13107 4.13303 8.89592 8.90013 8.90434 7.04 7.05 7.06 7.07 7.08 7.09 7.10 49.9849 50.1264 50.2081 2.65895 2.66083 2.66271 8.40833 8.41427 8.42021 353.393 351.895 3:56.101 1.91929 1.92019 1.92100 4.13498 4.13693 4.1.3887 8.90854 8.91274 8.91693 50.4100 2.06158 8.12015 357.911 1.92200 4.14082 8.92112 7.11 7.12 7.13 50,5521 50.6914 50.8369 2.00046 2.66833 2.07021 8.43208 8.43801 8.41393 359.425 360.94 302.407 1.92290 1.92380 1.92470 1.14276 4.14470 4.14664 8.92531 8.92949 8.93367 7.14 7.15 7.16 50.9796 51.1225 51.2656 2.07208 2.67395 2.67582 8.44985 8.45577 8.46108 303.991 365.520 367.002 1.92560 1.92650 1.92740 4.14858 4.15052 4.15245 8.93784 8.94201 8.94618 7.17 7.18 7.19 51.4089 51.5524 51.69961 2.07709 2.07955 2.68112 8.46759 8.47319 8.179:39 368.602 370.1.10 371.695 1.92829 1.92919 1.93008 4.15-138 4.15631 4.15827 8.95034 8.95450) 8.95806 7.20 51.8400 2.68328 8.4828 373.248 1.93098 4.16017 8.9281 7.21 7.22 7.23 51.9841 52.1284 52.2729 2.68511 2.68701 2.68887 8.40117 8.49706 8.50294 374.805 376.367 377.933 1.93187 1.93277 1.93366 4.10209 4.16402 4.10594 8.9668 8.97110 8.97524 7.24 7.25 7.20 52.4176 52.5625 52.7076 2.69072 2.09258 2.09444 8.50882 8.51409 8.52050 379.503 381.6078 382.657 1.93455 1.93547 1.93033 4.16786 4.10978 4.17169 8.97938 8.98351 8.98764 7.27 7.28 7.29 7.30 52.8529 52.9987 53.1411 2.69029 2.69815 2.7000) 8.52613 8.5:1229 8.7.3815 384.241 385,828 387.420) 1.93722 1.9:3810 1.9389) 4.17361 1.17502 1.17713 8.99176 8.99588 9.00000 53.29100) 2.70185 8.54 100 1.93988 7.31 7.32 7.33 53.43361 53.5824 53.7289 8.54985 8.75070 8.56151 389.017 390.618 392.223 393.833 1.94076 1.94163) 1.91253 4.17934 4.18125 4.18316 4.18506 9.00411 9.00822 9.01233 9.01643 2.70370 2.70555 2.707-10 2.70921 2.71109 2.71293 7.34 7.35 7.36 53.8756 54.0225 54.1696 8.56738 8.57321 8.57904 395.447 397.000 398.688 1.94341 1.94430 1.94518 4.18696 4.18886 4.19076 9.02053 9.02462 9.02871 54.3169 54.4641 54.6121 2.71477 2.71062 2.71816 8.58487 8.59069 8.5.2001 400.316 401.947 403.583 1.94006 1.9469-1 1.94782 4.19266 4.1955 4.19644 9.03280 9.03689 9.04097 7.37 7.38 7.39 7.40 7.41 7.42 7.43 51.7600 2.72029 8.00233 1.94870 4.19834 9.04504 54.98081 05.0564 55.2049 8.00814 8.61394 8.61974 405.224 406.809 408.518 410.172 1.94957 1.95045 1.95132 4.20023 4.20212 4.20400 9.04911 9.05318 9.05725 2.72213 2.72397 2.72080 2.72764 2.72147 2.73130 7.44 7.45 55.3536 55.5025 55.6516 8.62554 8.63134 8.63713 1.98220 1.95307 1.95395 4.20589 4.20777 4.20905 17.46 411.831 413.4.4 415.101 416.833 418.509 420.190 9.06131 9.06537 9.06942 9.07347 9.07752 9.08156 11 7.47 7.48 7.49 55.8009 55.9504 56.1001 2.73313 2.73496 2.73679 8.64292 8.64870 8.65148 1.95482 1.95509 1.95656 4.21153 4.21341 4.21529 ·xxü Powers and Roots (11 n na V10 n n8 10 n 100 n 7.50 56.2500 2.73861 8.66025 421.875 1.95743 4.21716 9.08560 7.51 7.52 7.53 56.4001 56.5504 56.7009 2.74044 2.74226 2.74408 8.66603 8.67179 8.67756 423.565 425.259 426.958 1.95830 1.95917 1.96004 4.21904 4 22091 4.22278 9.08964 9.00367 9.09770 - 7.54 7.55 7.56 56.8516 57.0025 57.1536 8.68332 8.08.907 8.69483 1.26091 1.46177 1.96264 4.22465 4.22651 4.22838 9.10173 9.10.575 9.10977 2.74591 2.74773 2.74955 2.75136 2.75318 2.75500 7.57 7.58 7.59 57.3019 57.4564 57.6081 428.661 430.369 432.081 433.798 435.520 437.245 438.976 410.711 442.451 444.195 1.96350 1.96437 1.6523 9.11378 9.11779 9.12180 7.60 7.61 7.02 7.63 1.96610 8.70057 8.70632 8.71206 8.71780 8.72353 8.72926 8.73199 57.7600 57.9121 58.0044 38.2169 58.3696 58.5225 58.6756 2.75681 2.75862 2.70043 2.76225 4.23024 4.23210 4.23396 4.23582 4.23768 4.23954 4.24139 1.9669M 1.96782 1.96808 9.12581 9.12981 9.13380 9.13780 7.64 7.05 7.66 2.76105 2.76586 2.76767 8.74071 8.74613 8.70214 445.944 417.697 449.455 1.96951 1.97040 1.97126 4.24324 4.24509 4.24694 9.14179 9.14577 9.14976 9.15374 9.15771 9.10109 7.67 7.08 7.09 7.70 58.8289 58.9824 59.1361 2.76948 2.77128 2.77308 8.75785 8.76356 8.70920 451.218 452.985 454.757 1.97211 1.97297 1.97383 4.24879 4.25063 4.27248 B2.200 2.77489 8.77416 456.5.33 1.97408 4.25432 9.10500 59.4441 59.5984 59.7529 2.77069 2.77849 2.78029 8.78066 8.78038 8.79204 458.314 460.100 461.890 4.25010 4.25800 4.25984 9.16962 9.17359 9.17754 1.97551 1.97639 1.97724 1.97809 1.97895 1.97980 59.9076 60.0625 60.2176 2.78209 2.78388 2.78568 463.685 405.484 407.289 4.26167 4.26351 4.20534 9.18150 9.18515 9.1820 7.71 7.72 7.73 7.74 7.75 7.76 7.77 7.78 7.79 7.80 7.81 7.82 7.83 8.79773 8.80311 8.80909 8.81476 8.82043 8.82010 60.3729 60.5284 60.6841 60.8400 2.78747 2.78927 2.79106 2.79285 469.097 470.911 472.729 474.552 1.98065 1.98150 1.982:34 4.26717 4.20100 4.27083 1.98319 4.27266 9.19335 9.19729 9.20123 9.20516 9.20910 9.21302 9.21695 8.83176 8.83742 8.84308 8.84873 2.79164 2.79643 2.79821 4.27448 4.27031 4.27813 60.9961 61.1524 61.3089 61.4656 61.0225 61.7796 476.380 478.212 480.019 481.890 483.737 485.588 1.98104 1.98189 1.98573 1.98058 1.98742 1.98826 2.80000 2.80179 2.80357 8.85438 8.80002 8.86566 4.27995 4.28177 4.28359 9.22087 9.22479! 9.22871 9.23202 9.23653 9.24043 61.9369 62.091 62.2021 2.80535 2.80713 2.80891 8.87130 8.87694 8.88257 487.443 481.304 491.169 1.98911 1.98995 1.99079 4.28540 4.28722 4.28903 62.4100 2.81069 8.88819 493.0:39 1.99163 4.29081 9.2443+ 7.84 7.85 7.86 7.87 7.88 7.89 1790 7.91 7.92 7.93 7.94 7.95 7.96 7.97 7.98 7.99 2.81247 2.81425 2.81603 8.89382 8.89944 8.90505 1.99217 1.99331 1.99415 4.29265 4.201446 4.29627 9.24823 9.25213 9.25602 62.5681 62.7264 62.8849 63.0436 63.2025 63.3616 494.914 49.793 498.677 500,566 502.400 504.358 2.81780 2.81957 2.82135 8.91067 8.91028 8.92188 1.99499 1.99582 1.99666 4.29807 4.29987 4.30108 4.30348 4.30528 4.30707 9.25991 9.20380 9.26708 9.27156 9.27544 9.27931 63.5209 63.6804 63.8401 2.82312 2.82189 2.82666 8.92749 8.93308 8.93868 106.262 08.170 510.082 1.99750 1.99833 1.99917 ·II) Powers and Roots xxiii n na vn ✓10n yn V10 n $100 n 8.00 64.0000 2.82843 8.94427 512.000 2.00000 9.28318 8.01 8.02 8.03 64.1601 64.3204 64.4809 2.83019 2.83136 2.83373 8.97986 8.95545 8.96103 513.922 515.850 517.782 2.00083 2.00107 2.00250 4.30887 4.31060 4.31246 4.31425 9.28704 9.2891 9.29477 8.04 8.05 8.06 64.6416 64.8025 61.9636 519.718 521.660 523.607 4.31604 4.31783 4.31961 9.29862 9.30248 9.30633 8.96660 8.97218 8.97775 8.98332 8.08888 8.99444 8.07 8.08 8.09 8.10 2.83549 2.83725 2.83901 2.84077 2.84253 2.84429 2.84605 65.1249 65.2864 65.4481 525.558 527.514 529.475 2.00333 2.00116 2.00199 2.00582 2.00664 2.00747 2.00830 2.00912 2.00905 2.01078 4.32140 4.32318 4.32497 4.32075 9.31018 9.31402 9.31786 9.32170 05.6100 9.00000 65.7721 65.9314 66.0969 2.84781 2.84956 2.85132 9.00555 9.01110 9.01663 531.441 533.412 535.387 537.308 4.32853 4.33031 4.33208 8.11 8.12 8.13 8.14 8.15 8.16 9.32553 9.32936 9.33319 9.33702 9.34084 9.34466 66.2496 66.4225 66.5856 2.85307 2.85182 2.85657 9.02219 9.02774 9.03327 539.353 511.343 643.338 2.01160 2.01242 2.01325 4.33386 4.33563 4.33741 8.17 8.18 8.19 66.7489 66.9124 67.0761 2.85832 2.80007 2.80182 2.86356 9.03881 9.01434 9.04980 5155.339 577.343 519,353 2.01407 2.0189 2.01571 4.33918 4.34095 4.34271 9.34847 9.35229 9.35610 8.20 67.2400 9.05539 551.308 2.01053 4.34448 9.35990 8.21 8.22 8.23 67.4041 67.5081 67.7329 2.86531 2.80705 2.86880 5:33.388 555.412 557.442 2.01735 2.01817 2.01899 4.31025 4.34801 4.34977 9.06091 9.00042 9.07193 9.07744 9.08295 9.08845 9.36370 9.36751 9.37130 9.37010 9.37889 9.38268 8.24 8.25 8.26 67.8976 08.0625 68.2276 2.87054 2.87228 2.87402 559.476 561.016 503.560 2.01980 2.02002 2.02144 4.35153 4.33329 4.35505 8.27 8.28 8.29 8.30 68.3929 68.5584 68.7241 2.87076 2.87750 2.87924 9.09395 9.09945 9.10404 565.609 567.061 109.723 2.02225 2.02307 2.02388 4.35081 4.35856 4.30032 9.38646 9.39024 9.39402 68.8900 2.88097 9.11043 571.787 2.02 169 4.36207 9.3.780 8.31 8.32 8.33 09.0561 09.2224 69.3889 2.88271 2.88114 2.88617 9.11592 9.12140 9.12088 573.850 575.930 578.010 2.02551 2.02632 2.02713 4.30382 4.36557 4.36732 8.34 8.35 8.36 69.5556 69.7225 69.8896 2.88791 2.885661 2.89137 9.13230 9.13783 9.11:330 580.094 582.183 584.277 2.02794 2.02875 2.02956 4.36907 4.37081 4.37256 9.40157 9.40534 9.40911 9.41287 9.41663 9.42039 9.42414 9.42789 9.43164 70.0.569 70.2244 70.3921 2.89310 2.89482 2.899655 9.14877 9.16423 9.1599 586.376 588.480 53.590 2.03037 2.03118 2.03199 4.37430 4.37401 4.37778 8.37 8.38 8.39 8.40 8.41 8.42 8.43 70.5000 2.89828 9.10515 592.704 2.03279 4.37952 9.43539 70.7281 70.824 71.0649 2.90000 2.90172 2.90315 594.823 58.918 599.077 4.38126 4.38299 4.38473 9.17001 9.17606 9.18150 9.18095 9.19239 9.19783 2.033t;0 2.034-10 2.03521 2.03601 2.03082 2.03762 9.43913 9.44287 9.44661 9.45034 9.45407 9.45780 8.44 8.45 8.46 71,2336 71.4025 71.15716 2.90517 2.8089 2.90861 601.212 603.351 605.496 4.38646 4.38819 4.38992 8.47 8.48 8.49 71.7409 71.9104 72.0801 2.91033 2.91204 2.91:376 9.20326 9.20869 9.21412 607.645 009.800 611.960 2.03842 2.03.23 2.04003 4.39165 4.39338 4.39510 9.46152 9.46528 9.46897 ·XXIV Powers and Roots [II V10 n ns V10 n 100 n vñ 2.91548 8.50 72.2500 9.21954 614.125 4.39683 8.51 8.52 8.53 2.91719 2.91890 2.92062 9.22497 9.23038 9.23580 616.295 618.470 620.650 2.04083 2.04103 2.04243 2.04323 4.39855 4.10028 4.40200 9.47268 9.47640 9.48011 9.48381 72.4201 72.5904 72.7609 72.9316 73.1025 73.2736 8.54 8.55 8.86 2.92233 2.92404 2.92575 9.24121 9.24062 9.25203 022.836 025.020 027.223 2.04402 2.04182 2.04562 4.40372 4.405-13 4.40715 9.48752 9.49122 9.49492 8.57 8.58 8.59 8.60 73.4449 73.6164 73.7881 73.9600 2.927476 2.92910 2.93087 9.25743 9.26283 9.26823 029.423 031.629 633.810 2.01641 2.0 1721 2.04801 4.10887 4.41058 4.41229 636.050 2.04880 4.41400 8.01 8.62 8.63 74.1321 74.3014 74.4769 2.93258 2.93128 2.935.98 2.93769 2.93939 2.91109 2.94279 9.27362 9.27X01 9.28410 9.28978 9.29.716 9.30054 9.30591 2.01959 2.0.039 2.05118 638.277 040.501 6+2.736 644.973 647.215 619.402 4.41571 4.41742 4.41913 4.42084 4.42074 4.42425 9.49861 9.50231 9.50000 9.50969 9.51337 9.51705 9.52073 9.52411 9.52808 9.53175 9.53542 : 9.53208 9.54274 8.64 8.65 8.66 74.6496 74.8220 74.9956 2.05197 2.05276 2.05305 8.67 8.68 8.69 75.1689 75.3424 75.5101 2.14449 2.94618 2.4788 9.31128 9.31605 9.32202 051.714 653.972 656.235 2.05434 2.05513 2.05592 4.42595 4.42765 4.42935 8.70 75.6900 2.94958 9.32738 678.503 2.05671 4.43105 9.54640 8.71 8.72 8.73 75.8611 76.0384 76.2129 2.95127 2.95296 2.95466 9.3:3274 9.33809 9.31315 660.776 063.055 665.339 2.05750 2.05828 2.05907 4.43274 4.43444 4.43013 9.55006 9.55371 9.05736 8.74 8.75 8.76 76.3876 76.5625 76.7376 2.95035 2.95801 2.95073 9.31880 9.354114 9.35919 667.628 (69.922 072,221 2.05986 2.00064 2.06143 4.43783 4.43952 4.44121 9.56101 9.56466 9.56830 1 8.77 8.78 8.79 76.9129 77.0884 77.2641 2.96142 2.96311 2.9479 9.30483 9.37017 9.37550 074.526 676.830 679.151 2.06221 2.06299 2.06378 4.44290 4.44459 4.44627 9.57194 9.575571 9.57921 8.80 77.4100 2.96048 9.38083 081.472 2.06156 4.4479 9.78284 8.81 8.82 8.83 2.96816 2.86985 2.97153 9.38616 9.39149 9.399081 683.798 086.129 688.465 2.00534 2.00612 2,06690 4.41814 4.40133 4.45301 2.58647 9.2009 9.5.9372 77.6161 77.7924 77.9689 78.1450 78.3225 78.4996 8.84 8.85 8.86 2.97321 2.97189 2.97658 9.40213 9.40744 1.41276 690.807 693.154 095.506 2.06768 2.06846 2.06924 4.45469 4.45637 4.45805 9.59734 9.60095 9.60407 8.87 8.88 8.89 78.6769 78.8744 79.0321 2.97825 2.97993 2.98161 9.41807 9.42338 9.42868 0977.864 700.227 702.595 2.07002 2.07080 2.07157 4.45972 4.46140 4.46307 9.60818 9.61179 9.61540 8.90 79.2100) 2.98329 9.43398 704.969 2.07237 4.46475 9.61.900 79.3881 79.5664 79.7449 2.9849 2.98064 2.98831 707.318 709.732 712.122 2.07313 2.073:10 2.07468 4.46642 4.46809 4.40976 9.62260 9.62620 9.02980 8.91 8.92 8.93 8.94 8.95 8.96 9.43928 9.14458 9.44987 9.45516 9.4604 9.46573 2.98998 2.99166 2.99333 714.517 716.917 719.323 79.9236 80.1025 80.2816 80.4609 80.6404 80.8201 2.07545 2.07622 2.07700 2.07777 2.07854 2.07931 4.47142 4.47309 4.47476 4.47642 4.47808 4.47974 9.63339 9.63698 9.64057 9.64415 9.64774 9.65132 8.97 8.98 8.99 2.99500 2.9%666 2.99833 9.47101 9.47629 9.48176 721.734 724.151 726.573 ·11] Powers and Roots XXV n n2 22_ vn vion ni 10 n V100 n 81.0000 3.00000 9.48683 729.000 2.08008 4.48140 9.63489 9.00 9.01 9.02 9.03 81.1801 81.3004 81.0409 3.00167 3.00333 3.00500 9.49210 9.457:37 9.50203 731.433 733.871 736.314 2.OS085 2.08162 2.08239 4.48306 4.48472 4.48638 9.65817 9.66204 9.66561 9.04 9.05 9.06 81.7216 81.9025 82.0836 3.00666 3.00832 3.00998 9.50789 9.51315 9.51840 738.703 741.218 743.677 2.08316 2.08:303 2.08-170 4.18803 1.48139 4.19134 9.60918 9.67274 9.67630 9.07 9.08 9.09 82.2649 82.4404 82.6281 3.01164 3.01330 3.01496 9.52305 9.528:30 9.53415 746.143 7-18.613 751.089 2.08546 2.08623 2.08099 4.49299 4.49.164 4.4729 9.07986 9.68342 9.68697 9.10 S.SI.() 3.011 73.3...11 2.04773 4.1:17:11 9.11.17 9.11 9.12 9.13 82.9921 83.1764 83.3569 3.01828 3.01993 3.02159 9.54403 9.5-1987 9.5.5.510 756.058 758.551 761.018 2.08852 2.08929 2.08005 4.49959 4.50123 4.50288 9.0407 9.09762 9.70116 9.14 9.15 9.16 83.5396 83.7225) 83.9006 3.02324 3.02490 3.02055 9.50033 9.50556 9.57079 763.552 700.061 708.575 2.09081 2.00108 2.09234 4.50452 4.50016 4.50781 9.70170 9.70824 9.71177 9.17 9.18 9.19 84.0889 84.2724 84.4561 3.02820 3.02985 3.03150 9.57001 9.58123 9.18045 771.095 773.621 776.1.52 2.09310 2.09380) 2.09102 4.509.135 4.51108 4.51272 9.71531 9.71884 9.722:36 84.6400 3.03315 9.59160 2.09.5:38 4.51430 9.72589 9.20 9.21 9.22 9.23 81.8211 85.008- 85.1929 3.03480 3.03645 3.03809 9.5.X687 9.60208 9.60729 778.688 781.2:30 783.777 786.3:30 2.0314 2.08600 2.09765 4.51599 4.51763 4.51926 9.72141 9.73203 9.73645 9.24 9.25 9.26 85.3776 85.5025 85.7476 3.03974 3.04138 3.04302 9.01249 9.61709 9.02289 788.889 7.1.453 794.023 2.09841 2.09917 2.09992 4.52089 4.52252 4.52415 9.73996 9.74348 9.74699 9.27 9.28 9.29 85.9329 80.1181 80.3041 3.0167 3.04631 3.04795 9.02808 9.03328 9.6.3810 7X6.598 799.179 801.705 801.357 2.10068 2.10144 2.10219 4.52578 4.52740 4.52:03 9.75049 9.75100 9.75750 9.30 86.49% 3.01939 9.61365 2.10294 4.53000 9.76100 9.31 9.32 9.33 80.6761 80.8624 87.0489 806.954 809.08 812.106 2.10:370) 2.10445 2.10520 4.53228 4.533) 4.53552 9.76450 9.76799 9.77148 3.05.23 3.05387 3.05450 3.05611 3.0,7778 3.05941 9.64883 9.05401 9.65919 9.66437 9.66.54 9.67471 9.34 9.35 9.36 87.2356 87.4225 87.0096 814.781 817.400 820.020 2.10595 2.10671 2.107 10 4.53714 4.53876 4.51038 9.77497 9.77846 9.78195 9.37 9.38 9.39 87.7909 87.0844 88.1721 3.06105 3.06208 3.003131 9.07988 9.08504 9.69020 822.677 821.294 827.9836 2.10821 2.108% 2.10971 4.54199 4.54361 4.51522 9.78543 9.78891 9.79239 88.3000 3.6594 9.09.30 8:30.584 2.110-15 4.54684 9.79586 9.40 9.41 9.42 9.43 9.44 9.45 9.46 88.5181 88.7304 88.9249 3.06757 3.00920 3.07083 9.70052 9.70567 9.71082 833.238 835.897 838.562 4.54845 4.55006 4.55167 2.11120 2.11195 2.11270 2.11344 2.11419 2.11494 9.79933 9.80280 9.80627 9.80974 9.81320 9.81666 9.71597 9.72111 9.72625 4.55328 4.55488 4.55649 89.1136 89.3025 89.4916 89.6809 89.8704 90.0601 3.07246 3.07409 3.07571 3.07734 3.07896 3.08058 841.232 813.909 846.591 849.278 851.971 851.670 9.47 9.48 9.49 9.73139 9.73653 9.74166 2.11568 2.11642 2.11717 4.55809 4.55970 4.56130 9.82012 9.82357 9.82703 ·XXVI Powers and Roots III n 22 vn ✓10 n no yn 10 n 100 n 90.2600 3.08221 9.74679 857.375 2.11791 4.56290 9.83048 9.50 9.51 9.62 9.53 90.4401 90.6304 90.8209 3.08383 3.08545 3.08707 9.75192 9.75705 9.76217 800.085 862.801 865.523 2.11865 2.11940 2.12014 4.56450) 4.36610 4.56770 9.83392 9.83737 9.84081 9.54 9.55 9.56 91.0116 91.2025 91.3936 3.08869 3.09031 3.09192 9.76729 9.77241 9.77753 868.251 870.984 23 2.12088 2.12102 4.56930 4.57089 4.57249 9.84425 9.84769 113 9.57 9.58 9.59 91.5819 91.7764 91.9081 3.09354 3.09716 3.0277 9.78264 9.78775 9.79285 876.467 879.218 881.974 2.12310 2.12384 2.12458 4.57408 4.57567 4.57727 9.85456 9.85799 9.86142 9.60 92.1600 3.09839 9.79796 884.736 2.12532 4.07886 9.86485 92.3521 92.5144 92.7369 3.1000 3.10161 3.10322 9.80306 9.80816 9.81326 887.504 890.277 893.050 2.12605 2.12079 2.12753 4.58045 4.58204 4.58362 9.61 9.62 9.63 9.64 9.65 9.66 9.868271 9.87169 9.87511 9.87853 9.88195 9.88536 92.9296 93,1225 93.3156 3.10483 3.10014 3.10805 9.81835 9.82344 9.82853 895.841 898.632 901.429 2.12820 2.12000 2.12974 4.58521 4.58679 4.58838 9.07 9.68 9.69 3.10966 3.11127 3.11288 9.83362 9.83870 9.84378 94.231 907.0:39 99.873 2.13047 2.13120 2.13194 4.589X 1.59151 4.59312 9.88877 9.89217 9.89558 9.70 93.5089 93.7024 93.8X1 94.090 94.2841 94.4784 94.6729 2.13267 4.59470 9.89898 3.11448 3,11609 3.11769 3.11929 9.81886 9.85393 9.85.901 9.80408 912.673 915.499 918.3:30 921.167 2.13310 2.1:3414 2.13487 9.90238 9.90578 9.90918 9.71 9.72 9.73 9.74 9.75 9.76 4.5928 4.59780 4.59943 4.60101 4.60258 4.60416 94.8676 95.0625 95.2576 3.12000 3.12250 3.12410 9.86914 9.87-421 9.87927 924.010 926.859 929.714 2.1360 2.13633 2.13700 9.91257 9.91596 9.91935 9.77 9.78 9.79 95.4529 95.6184 95.8441 3.12070 3.12730 3.12890 9.88433 9.88939 9.89444 932.575 935.441 938.314 2.13779 2.13852 2.13925 2.13997 4.60573 4.60730 4.60887 4.61044 9.92274 9.92612 9.92950 9.80 96.0400 3.13050 9.89949 9.93288 9.81 9.82 9.83 96.2361 96.4324 96.6289 3.13209 3.13369 3.13528 9.90452 9.:X0959 9.91464 941.192 944.076 916.80 949.862 2.14070 2.14143 2.14216 4.61200 4.61357 4.61514 9.93626 9.93964 9.94301 9.84 9.85 9.86 96.8256 97.0225 97.2196 3.13688 3.13817 3.14006 9.91968 9.92472 9.92975 952.764 955.672 958.585 2.14288 2.14361 2.14433 4.61670 4.61826 4.61983 9.94638 9.94975 9.95311 9.87 9.88 9.89 9.90 97.4169 97.6144 97.8121 98.0100 3.14166 3.14325 3.14481 9.93479 9.93982 9.91485 961.505 964.4:30 967.302 2.14506 2.14578 2.11651 4.62139 4.62295 4.62451 9.91648 9.95984 9.96320 3.14043 9.94987 970.299 2.14723 4.62607 9.96655 9.91 9.92 9.93 3.14802 3.140X0 3.15119 9.954.90 9.95992 9.96494 973.242 976.191 979.147 2.14795 2.14867 2.1440 4.02702 4.62918 4.63073 9.96991 9.97326 9.97661 98.2081 98.4064 98.0049 98.80365 99.0025 99.2016 9.94 9.95 9.96 3.15278 3.15436 3.15095 9.96995 9.97497 9.97998 982.108 985.075 988.048 2.15012 2.15084 2.15156 4.63229 4.63384 4.63539 9.97996 9.98331 9.98665 9.97 9.98 9.99 99.4009 99.6004 99.8001 3.15753 3.15911 3.16070 9.98499 9.98999 9.99500 991.027 994.012 997.003 2.15228 2.15300 2.15372 4.63694 4.63849 4.61004 9.98999 9.99333 9.99667 ·TABLE III IMPORTANT NUMBERS A. Units of Length ENGLISTI UNITS METRIC UNITS 1 centimeter (cm.) 12 inches (in.) = 1 foot (ft.) 3 feet 1 yard (yd.) 64 yards 1 rod (rd.) 320 rods 1 mile (mi.) 10 millimeters (mm.) 10 centimeters 10 decimeters 10 meters 1000 meters 1 decimeter (dm.) =1 meter (11.) 1 dekameter (Dm.) 1 kilometer (Km.) ENGLISIT TO METRIC METRIC TO INGLISH 1 cm. 1 in. 1 ft. 1 mi. = 2.5400 cm. = 30.430 ein. = 1.6093 Kin. 0.3937 in. 39.37 in. : 0.0214 ini. . 1 m. 3.2808 ft. 1 Km. B. square yard 1 acre (A.) 1 square mile Units of Area or Surface 9 square feet 1290 square inches 100 square rods : :18-10 squire yards 640 acres 102400 square rods C. Units of Measurement of Capacity DRY MEASURE LIQUID MEASURE 2 pints (pt.) 1 quart (qt.) 4 gills (gi.) 1 pint (pt.) 8 quarts 1 peck (pk.) 2 pints 1 quart (qt.) 4 pecks = 1 bushel (bu.) 4 quarts 1 gallon (gal.) 1 gallon 231 cu. in. D. Metric Units to English Units 1 liter 1000 cu. cm. - 01.02 cu. in. 1.0567 liquid quarts 1 quart .94030 liter 110.36 cu. cm. 1000 grams 1 kilogram (kg.) = 2.2046 pounds (lb).) 1 pound .45:3593 kilograin 453.59 grams E. Other Numbers Tratio of circumference to diameter of a circle 3.14159205 1 radian angle subtended by an arc equal to the radius : 57° 17' 44'.8 = 570.2957796 = 180°/ 1 degree = 0.01745320 radian, or 7/180 radians Weight of 1 cu. ft. of water = 62.425 lb. vxvii ·SYMBOLS AND ABBREVIATIONS The following symbols and abbreviations are used for the sake of brevity throughout the present book : = equal, or is equal to + not equal, or is not equal to > greater than < less than is congruent to I perpendicular, or is perpendicular to 11 parallel, or is parallel to similar, or is similar to Zangle angles A triangle A triangles O parallelogram S parallelograms O circle circles Ax. Axiom Cons. Construction, or by construction Cor. Corollary Def. Definition Hyp. Ilypothesis, or by hypothesis Iden. being identical Prop. Proposition rt. right st. straight Th. Theorem Prob. Problem Fig. Figure or diagram and so on hence or therefore arc The signs +, -, X, *, are used with the same meanings as in algebra. The following agreements are also made : axb=a.b = ab, a = b = a/b = a;b xxviii ·SYLLABUS OF PLANE GEOMETRY INTRODUCTION PART I. DRAWING SIMPLE FIGURES PART II. FUNDAMENTAL IDEAS PART III. STATEMENTS FOR REFERENCE 27. Axioms. a c ALE 1. If equals are added to equals, the sums are equal. Thus, if a= b and c = :d, then a + c = Eltd. 2. If equals are subtracted from equals, the remainders are equal. Thus, if a = b and c= d, theni ar - C=-- d. 3. If equals are multiplied by equals, the products are equal. Thus, if a = b and c :, then ac od. 4. If equals are divided by eqnals, the quotients are equal. Thus, if 0 b and ca dl, then iz In applying this axiomn it is supposed that cand il are not equal to zero. 5. If equals are added to unequals, the results are unequal and in the same order. Thus, if a b and c>d, then a + c>b + d. 6. If equals are subtracted from unequals, the results are unequal and in the same order. Thus, if a>h and c = 0, then a -6>Ud. 7. If unequals are added to unequals in the same sense, the results are unequal in the same order. Thus, if a >b and c>d, then a +0> b+d. 8. If unequals are subtracted from equals, the results are unequal in the opposite order. Thus, if a = b and c>d, then a-c<b-d. 9. Quantities equal to the same quantity, or to eqiial quantities, are equal to each other. In other words, a quantity may be substituted for its equal at any time in any expression. * This syllabus contains the principal statements of fact, with their article oumbers, from the Plane Geometry by the authors of this book. xxix ·XXX SYLLABUS OF PLANE GEOMETRY 10. The whole of a quantity is greater than any one of its parts. 11. The whole of a quantity is equal to the sum of its parts. 28. Postulates. 1. Only one straight line can be drawn joining troo given points. 2. A straight line can be extended indefinitely. 3. A straight line is the shortest curve that can be drawn between two points. 4. A circle can be described about any point as a center and with a radius of any length. 5. A figure can be moved unaltered to a new position. 6. All straight angles are equal. Hence, also, all right angles are equal, for a right angle is half of a straight angle. 30. Preliminary Construction Problems. 1. To construct a triangle, each of whose sides is equal to a given length. $ 3. 2. To construct a triangle, whose three sides are, respectively, equal to three given lengths. $ 4. 3. To construct a perpendicular to a given straight line at a given point in that line. $ 5. 4. To construct a perpendicular to a given line from a given point not on that line. $ 6. 5. To construct, at a given point in a given line, another line that makes an angle equal to a given angle with the given line. $7. 6. To divide a portion of a straight line into two equal parts. (To bisect a line.) § 8. 7. To divide a given angle into two equal parts. (To bisect an angle.) $ 9. 31. Preliminary Theorems. 1. All radii of the same circle are equal. $ 2; and $ 23. 2. Circles whose radii are equal can be placed upon each other so that their centers and their circumferences coincide (lie exactly upon each other). 3. Equal angles may be placed upon each other so that their vertices coincide and their corresponding sides fall along the same straight lines, This is, in fact, what we mean by equal angles.

·SYLLABUS OF PLANE GEOMETRY xxxi 4. Two straight lines have at most one point in common. See postulate 1, § 28. 5. Two circles have at most two points in common. See § 8. 6. A straight line and a circle may have at most two points in comnion. 7. At a given point in a given line only one perpendicular can be drawn to that line. (A consequence of Problern :3, $ 5.) 8. Complements of the same angle, or of equal angles, are equal. 9. Supplements of the same angle, or of equal angles, are equal. 10. Vertical angles are equal. § 19. 11. If two adjacent angles have their exterior sides in a straight line, they are supplementary. $ 18. 12. If two adjacent angles are supplementary, they have their exterior sides in a straight line. 13. If cach of two figures can be placed upon a third figure so as to coincide with it, they can be placed upon each other so that they coincide. 14. Any desired angle may be drawn, and any angle may be measured, by the use of a protractor. (But the use of this instrument is not permitted when a figure is to be constructed. See $ 20.) 15. A perpendicular to a given line through any given point may be drawn by means of a set square or a drawing triangle. (But the use of these instruments not permitted when a figure is to be constructed. See § 20.) 16. The area of a rectangle (in terms of a unit square) is equal to the product of its width and its height, measured in units of length equal to one side of the unit square. 17. The area of any given figure is greater than the area of any figure that is drawn completely within it. 18. The areas of two figures are equal if they consist of corresponding portions that can be made to coincide.

·CHAPTER I RECTILINEAR FIGURES PART I. TRIANGLES 35. Theorem I. If two triangles have two sides and the included angle of the one equal, respectively, to two sides and the included angle of the other, the triangles are congruent. 36. Corollary 1. Two right triangles are congruent if the two sides of the one are equal, respectively, to the two sides of the other. 37. Theorem II. If two triangles have two angles and the included side in the one equal, respectively, to two angles and the included side in the other, the triangles are congruent. 38. Corollary 1. Two right triangles are congruent if an acute angle and its adjacent side in one are equal, respectively, to an acute angle and its adjacent side in the other. 40. Theorem III. In an isosceles triangle the angles opposite the equal sides are equal. 41. Corollary 1. If a triangle is equilateral, it is also equiangular. 43. Theorem IV. The bisector of the angle at the vertex of an isosceles triangle is perpendicular to the base and bisects the base. 44. Corollary 1. In any isosceles triangle (a) The bisector of the angle at the vertex divides the triangle into two congruent right triangles. (6) The bisector of the vertical angle coincides with both the altitude and the median drawon through the vertex. (c) The perpendicular bisector of the base passes through the vertex, and divides the triangle into two congruent right triangles. 45. Theorem V. If two triangles have three sides of the one equal, respectively, to the three sides of the other, they are congruent. 46. Corollary 1. Three sides determine a triangle; that is, if the . three sides are given, the triangle is thereby fixed. 47. Theorem VI. An exterior angle of a triangle is greater than either or the opposite interior angles. xxxii ·SYLLABUS OF PLANE GEOMETRY xxxiii PART II. PARALLEL LINES 49. Parallel Postulate. Only one line can be drawn through a given point parallel to a given line. 50. Corollary 1. Lines parallel to the same line are parallel to each other. 51. Theorem VII. When two lines are cut by a transversal, if the alternate interior angles are equal, the two lines are parallel. 52. Corollary 1. Lines perpendicular to the same line are parallel. 54. Theorem VIII. If two parallel lines are cut by a transversal, the alternate interior angles are equal. 56. Theorem IX. If two lines are cut by a transversal and the corresponding angles are equal, the lines are parallel. 57. Corollary 1. If two lines are cut by a transversal and the two interior angles on the same side of the transversal are supplementary, the lines are parallel. 58. Corollary 2. From a given point only one perpendicular can be drawn to a given line. 59. Theorem X. (Converse of Theorem IX.) If two parallel lines are cut by a transversal, the corresponding angles are equal. 60. Corollary 1. If a line is perpendicular to one of two parallels, it is perpendicular to the other also. 61. Problem 1. To construct a line parallel to a given line and passing through a given point. PART III. ANGLES AND TRIANGLES 62. Theorem XI. The sum of three angles of any triangle is equal to two right angles, or 180°. 63. Corollary 1. The sum of the two acute angles of any right triangle is one right angle, or 90°. 64. Corollary 2. An exterior angle of any triangle is equal to the sum of its opposite interior angles. 65. Corollary 3. Each angle of an equilateral triangle is equal to 600.

·Xxxiv SYLLABUS OF PLANE GEOMETRY 66. Corollary 4. If two angles of one triangle are equal, respectively, to two angles of another triangle, then the third angles are likewise equal. 67. Theorem XII. Theo angles whose sides are respectively parallel are either equal or supplementary. 68. Theorem XIII. Tuo (ingles whose sides are respectively perpendicular to each other are either equal or supplementary. 69. Theorem XIV. Two right triangles are congruent if the hypotenuse and an acute angle of the one are equal, respectively, to the hypotenuse and an acute angle of the other. 70. Theorem XV. Two right triangles are congruent if the hypotenuse and a side of the one are equal, respectively, to the hypotenuse and a side of the other. 71. Corollary 1. If two oblique lines of equal length are drawn from a point C in a perpendicular CD to a line AB, they cut 0.1* equal distances from the foot of the perpendicular, and conversely. 72. Theorem XVI. (Converse of Theorem III.) If two angles of a triangle are equal, the sides opposite are equal, and the triangle is isosceles. 73. Corollary 1. An equiangular triangle is also equilateral. 74. Corollary 2. If tuo oblique lines are drawn from a point C in a perpendicular CD to a line AB, so as to make equal angles with AB, they are equal, and conversely. 75. Theorem XVII. If tuo angles of a triangle are unequal, the sides opposite them are'unequal and the greater side is opposite the greater angle. 76. Corollary 1. If two triangles have two sides of the one equal to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second. 77. Corollary 2. If two oblique lines are drawn from a point C in a perpendicular CD to a line AB, and if the base angles at A and B are unequal, the oblique line opposite the greater base angle is the greater; in particular, the perpendicular CD is itself the shortest line from C to any point of AB.

·SYLLABUS OF PLANE GEOMETRY XXXV 78. Theorem XVIII. If t100 sides of a triangle are unequal, the angles opposite them are unequal and the greater angle is opposite the greater side. 79. Corollary 1. If two triangles have two sides of the one equal to two sides of the other, but the third side of the first greater than the third side of the second, then the included angle of the first is greater than the included angle of the second. 80. Corollary 2. If from a point C in a perpendicular CD to a line AB unequal oblique lincs are drawn to the base AB, the longer of the oblique lines is opposite the larger of the two base angles. PART IV. QUADRILATERALS 82. Theorem XIX. Either diagonal of a parallelogram divides it into two congruent triangies. 83. Corollary 1. Any side of a parallelogram is equal to the side opposite it. 84. Corollary 2. The segments of parallel lines included between parallel lines are equal. 85. Theorem XX. If a quadrilateral has both pairs of opposite sides equal, it is a parallelogram. 86. Theorem XXI. If a quadrilateral has one pair of sides equal and parallel, it is a parallelogram. 87. Theorem xxli. The diagonals of a parallelogram bisect each other. 88. Theorem XXIII. Two parallelograms are congruent if two sides and the included angle of the one are equal, respectively, to two sides and the included angle of the other. 89. Theorem XXIV. The line joining the middle points of the two sides of a triangle is parallel to the base and equal to half the base. 90. Corollary 1. (Converse of $ 89.) The line drawn through the middle point of one side of a triangle parallel to the base bisects the other side.

·Xxxvi SYLLABUS OF PLANE GEOMETRY 91. Theorem XXV. If three parallel lines cut off two equal por. tions of one transversal, they cut off two equal portions of any other transversal. 92. Corollary 1. If a series of parallel lines cut of equal portions of one transversal, they cut off equal portions of any other transversal. 93. Corollary 2. If three parallel lines cut off two portions of one transversal, one of which is double the other, they cut ot two portions of any other transversal, one of which is double the other. 94. Corollary 3. If three parallel lines cut off tuo portions of one transversal, one of which is n times the other, they cut off two portions of any other transversal, one of which is n times the other. PART V. POLYGONS 97. Theorem XXVI. The sum of the interior angles of a polygon is two right angles taken as many times as the figure has sides, less two. 98. Theorem XXVII. The sum of the exterior angles of a polygon formed by producing the sides in succession is equal to four right angles. PART VI. THE LOCUS OF A POINT 100. Theorem XXVIII. The locus of all points equidistant from the extremities of a line is the perpendicular bisector of that line. 101. Theorem XXIX. The bisector of an angle is the locus of all points equidistant from its sides. 102. Supplementary Propositions on Altitudes, Medians, etc. Theorem XXX. The perpendiculars erected at the middle points of the sides of a triangle meet in a point. Theorem XXXI. The bisectors of the angles of a triangle meet in a point. Theorem XXXII. The altitudes of a triangle meet in a point. Theorem XXXIII. The medians of a triangle meet in a point which is two thirds of the distance from any vertex to the middle point of the opposite side.

·SYLLABUS OF PLANE GEOMETRY xxxvii CHAPTER II THE CIRCLE PART I. CIIORDS. ARCS. CENTRAL ANGLES 104. Postulates. (1) As a central angle increases, its intercepted arc increases, and vice versa; and as a central angle decreases, its intercepted are decreases, and vice versa. (2) In the same circle (or equal circles), equal central angles intercept equal arcs ; and equal tres subtend equal central angles. 106. Theorem I. In the same circle (or in equal circles), equal arcs subtend equal chords. 107. Theorem II. (Converse of $ 106.) In the same circles (or in cqual circles), equal chords subtend equal arcs. 108. Theorem III. A diameter perpendicular to a chord bisects the chord and the arc subtended by it. 109. Theorem IV. In the same circle (or in equal circles), equal chords are equally distant from the center, and, conversely, chords that are equally distunt from the center are equal. 110. Theorem V. In the same circle (or in equal circles), if trvo unequal chords are drawn, the longer one is ncarer the center. 111. Corollary 1. (Converse of $ 110.) In the same circle (or in equnl circles), if two chords are unequally distant from the center, the more remote is the less. PART II. TANGENTS AND SECANTS 115. Theorem VI. A line perpendicular to a radius at its extremity is tangent to the circle. 116. Corollary 1. A tangent to a circle is perpendicular to the radius drawn to the point of contact. 117. Corollary 2. A perpendicular to a tangent at its point of contact passes through the center of the circle. 118. Theorem VII. Two tangents drawn to a circle from a point outside are of equal length.

·Xxxvii SYLLABUS OF PLANE GEOMETRY 119. Theorem VIII. Two parullel lines intercept equal arcs on a circle. 120. Theorem IX. Through three given points not all on the same straight line, one and only one circle can be drawn. 121. Corollary 1. A circle may be drawn to circumscribe any triangle. 122. Corollary 2. The perpendicular visectors of the sides of a triangle meel in a point. 123. Corollary 3. Acircle may be completed if any arc of it is given. 124. Theorem X. A circle may be inscribed in any triangle. 125. Corollary 1. A circle drawn from any point on the visector of an angle, with a radius equal to the distance from that point to one side, is tangent to both sides of the angle. PART III. MEASUREMENT OF ANGLES 130. Theorem XI. In the same circle (or in equal circles), two central angles have the same ratio as their intercepted arcs. Note 1. We may assume as a postulate that if two geometric ratios are equal whenever their terms are commensurable, they are equal also when their terins are incommensurable. NOTE 2. A central angle is measured by its intercepted arc. 132. Theorem XII. An inscribed angle is measured by one half of its intercepted arc. 133. Corollary 1. Any angle inscribed in a semicircle is a right angle. 134. Corollary 2. Any angle inscribed in a segment greater than a semicircle is acute, while any angle inscribed in a segment less than u semicircle is obtuse. 135. Corollary 3. All angles inscribed in the same segment are equal. 136. Theorem XIII. An angle formed by two chords intersecting within a circle is measured by one half of the sum of the intercepted arcs. 137. Theorem XIV. An angle formed by a tangent and a chord drawn through the point of tangency is measured by one half of the intercepted arc.

·SYLLABUS OF PLANE GEOMETRY xxxix 138. Theorem XV. An angle formed by two secants, or by a tangent and a secant, or by two tangents that meet outside a circle is measured by one half the difference of the intercepted arcs. PART IV. CONSTRUCTION PROBLEMS 139. Problem 1. a circle. Through a given point to draw a tangent to 140. Problem 2. To circumscribe a circle about a given triangle. 141. Problem 3. To inscribe a circle in a given triangle. 142. Problem 4. On a given straight line to construct a segment of a circle that shall contain a given angle. CHAPTER III PROPORTION. SIMILARITY PART I. GENERAL THEOREMS ON PROPORTION 144. General Theorems on Proportion. Theorem A. In any proportion, the product of the extremes is equal to the product of the means. Corollary 1. If the two antecedents of a proportion are equal, the consequents are also equal. Theorem B. If the product of two numbers is equal to the product of two other numbers, either pair may be made the means of a proportion in which the other two are taken as the extremes. Theorem C. If your quantities are in proportion, they are in proportion by inversion. Theorem D. If four quantities are in proportion, they are in proportion by alternation. Theorem E. If four quantities are in proportion, they are in proportion by composition. Theorem F. If four quantities are in proportion, they are in proportion by division.

·xl SYLLABUS OF PLANE GEOMETRY Theorem G. Tf four quantities are in proportion, they are in proportion by composition and division. Theorem H. In a series of equal ratios the sum of the antecedents is to the sum of the consequents als any antecedent is to its consequent. PART II. PROPORTIONAL LINE-SEGMENTS 145. Theorem I. A line parallel to the base of a triangle divides the other sides proportionally. 146. Corollary 1. If a line is drawn parallel to the base of a triangle, either side is to one of its segments as the other side is to its corresponding segment. 147. Theorem II. (Converse of Theorein I.) If a line divides two sides of a triangle proportionally, it is parallel to the third side. 148. Corollary 1. If a line cuts two sides of a triangle in such a way that either side is to one of its segments as the other side is to its corresponding segment, then the line is parallel to the third side. 149. Theorem III. The bisector of an angle of a trianyle divides the opposite side into segments which are proportional to the sides of the angle. 150. Theorem IV. If a series of parallels be cut by two lines, the corresponding segments are proportional. 151. Problem 1. To divide a given line into parts proportional to any number of given lines. 153. Problem 2. To find the fourth proportional to three given lines. PART III. SIMILAR TRIANGLES AND POLYGONS 155. Theorem V. If two triangles are mutually equiangular, they are similar. 156. Corollary 1. Two triangles are similar if two angles of the one are equal respectively to two angles of the other. 157. Corollary 2. Two right triangles are similar if an acute angle of the one is equal to an acute angle of the other. 158. Theorem VI. (Converse of Theorem V.) If two triangles are similar, they are mutually equiangular.

·SYLLABUS OF PLANE GEOMETRY xli 159. Theorem VII. Two triangles are similar if un angle of the one equals an angle of the other and the including sides are proportional 161. Theorem VIII. If, in any right triangle, a perpendicular is drawn from the ertex of the right angle to the hypotenuse, the two right triangles thus formed are similar to each other and to the given triangle. 162. Corollary 1. In any right triangle the perpendicular from the vertex of the right angle to the hypotenuse is the mean proportional between the segments of the hypotenuse. 163. Corollary 2. If, in any right triangle, a perpendicular is drazon from the vertex of the right angle to the hypotenuse, each side of the right triangle is the mean proportional between the hypotenuse anil the segment adjacent to that side. 164. Problem 3. To find the mean proportional between two lines. 166. Theorem IX. Regular polygons of the same number of sides are similar. 167. Theorem X. The perimeters of two similar polygons are to each other in the same rutio (ls any two corresponding sides. PART IV. PROPORTIONAL PROPERTIES OF CHORDS, SECANTS, AND TANGENTS 168. Theorem XI. If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other. 169. Theorem XII. If from a point without a circle a secant and a tangent cure drawn, the tangent is the mean proportional between the entire secant and its exterior segment. 170. Theorem XIII. If from a fixed point without a circle any two secants are drawn, the product of one secant and its external segment is equal to the product of the other secant and its external segment. 172. Problem 4. To divide a given line segment in extreme and mean ratio.

·xlii SYLLABUS OF PLANE GEOMETRY PART V. SIMILAR RIGHT TRIANGLES. TRIGONOMETRIC RATIOS 174. Theorem XIV. If two right triangles have one acute angle of one equal to one acute angle of the other, their corresponding sides are in the same ratios. 175. Corollary 1. If an acute angle of a right triangle is known, the ratios of the sides are all determined. 176. Corollary 2. If the ratio of any pair of sides of a right triangle is given, the acute angles are determined. 179. Theorem XV. Corresponuliny altitudes divide any two similar triangles into two corresponding pairs of similar right triangles. 180. Corollary 1. Any two similar polygons may be subdivided into correspondiny pairs of similar right triangles. CHAPTER IV AREAS OF POLYGONS. PYTHAGOREAN THEOREM 181. Area of a Rectangle. The area of a rectangle is equal to the product of its base by its height. 182. Corollary 1. The area of a square is equal to the square of its side. 183. Corollary 2. The areas of two rectangles are to each other as the products of their bases and altitules. 184. Corollary 3. Two rectangles that have equal vltitudes (train, each other as their bases; two rectangles that have equal bases (ire to each other as their altitudes. 186. Theorem I. The area of a parallelogram is equal to the product of its base by its altitude. 187. Corollary 1. (a) Two parallelograms are to each other as the products of their bases and altitudes. (0) Two parallelograms that have equal bases and equal altitudes are equal in area.

·SYLLABUS OF PLANE GEOMETRY xliji 188. Corollary 2. Two parallelograms that have equal altitudes are to each other as their luses; two parallelograms that have equal bases are to each other as their altitulles. 189. Theorem II. The area of a triangle is equal to one half the product of its base by its altitude. 190. Corollary 1. (a) Two triangles are to each other as the products of their vases and altitudes. (1) Troo triangles that have equal bases are to each other as their altitudes. (c) Two triangles that have equal altitudes are to each other as their bases. (a) Tuo triangles that have equal bases and equal altitudes are equal in area. 191. Theorem III. The area of a trapezoid is equal to the product of its altitude and one half the sum of its buses. 192. Corollary 1. The area of a trapezoid is equal to the product of its altitude and the line joining the mill-points of the non-parallel sides. The area of a trapezoid is equal to the product of its altitude and its median. 193. Theorem IV. Two triangles that have an acute angle of the one equal to an cucute angle of the other are to each other as the products of the siles including the equal angles. 194. Theorem V. Similar triangles are to each other as the squares of any tro corresponding sides. 195. Corollary 1. The areas of 1200 similar polygons are to each other as the squares of any two corresponding sides. The areas of two similar polygons are to each other as the squares of any two corresponding lines, 196. Theorem VI. The Pythagorean Theorem. The square on the hypotenuse of a "right triangle is equivalent to the sum of the squares on the two sides. 197. Corollary 1. The square on either side of a right triangle is equivalent to the square on the hypotenuse diminished by the square on *he other side.

·xliv SYLLABUS OF PLANE GEOMETRY 199. Theorem VII. In any triangle the square on the side opposite an acute angle is equal to the sum of the squares on the other two sides diminished by twice the product of one of those sides and the projection of the other upon it. 200. Theorem VIII. In any obtuse triangle the square on the side opposite the obtuse angle is equal to the sum of the squares on the other sides increased by twice the product of one of those sides and the projection of the other upon it. 201. Problem 1. To construct a square whose area shall be equal to the sum of the areas of two given squares. 202. Problem 2. To construct a triangle whose area shall be equal to that of u given polyyon. CIIAPTER V REGULAR POLYGONS AND CIRCLES 204. Theorem I. If a circle is divided into a number of equal arcs: (a) the chords joining the points of division form a regular inscribed polygon; (1) tangents drawn at the points of division form a regular circumscribed polygon. 205. Theorem II. (a) A circle may be circumscribed about any regular polygon; (b) a circle may also be inscribed in it. 207. Theorem III. The area of a regular polygon is equal to half the product of its apothem and perimeter. 210. Areas and Lengths of Circles. If the number of sides of the regular inscribed and regular circumscribed polygons is repeatedly doubled : (a) their areas approach the area of the circle as a common limit; (0) their perimeters approach the length of the circumference of the circle as a common limit. 211. Theorem IV. The circumferences of two circles are to each other as their radii.

·SYLLABUS OF PLANE GEOMETRY xly -- 212. Corollary 1. The ratio of a circumference to its diameter is the same for all circles. 213. The Number TT. The number obtained by dividing the circumference of any circle by its diameter is denoted by the Greek letter . 214. Corollary 2. In any circle c = Trd, orc 2 Tr, where r is the radius, d the diameter, and a the length of the circumference. 215. Theorem V. The area of a circle is equal to one half the product of its radius and its circumference. 216. Corollary 1. The area of a circle is equal to 1 times the square of its radius, that is, A =72. 217. Corollary 2. The areas of low circles are to each other as the 8q7l/res of their ralii. 220. Problem 1. Given the siile amd radius of a regular inscribed polygon, to find the side of a regular inscribed polygon of double the number of sides. 221. Corollary 1. · 1, and s= the side of the inscribed polygon, the side AC of the regular inscribeul polygon of double the numher of siles is, AC=V2 222. Problem 2. To compute (.pprorimately the value of T. 0.28317 (approximately)= 3.14159 (usually written 3.1416, or 37). T 2 223. Problem 3. 224. Problem 4. 225. Problem 5. To inscribe a square in a yiven circle. To inscribe a regular hecagon in n given circle. To inscribe a regular decayon in a given circle. APPENDIX TO PLANE GEOMETRY MAXIMA AND MINIMA 226. Of all chords through P, the diameter PQ is the maximum (greatest). Of all regular polygons inscribed in a circle, the equilateral triangle has the minimum (least) (rea, or simply, is the minimum. Of all the straight lines that can be drawn from a fixed point to a given line, the perpendicular is the minimum.

·xlvi SYLLABUS OF PLANE GEOMETRY 227. Theorem I. Of all triangles that have the same two given sides, that in which these sites include a right angle is the maximum. 229. Theorem II. Of all isoperimetric triangles having the same base, the isosceles is the maximum. 230. Corollary 1. Of all isoperimetric triangles, the equilateral is the maximum. 231. Theorem III. Of all isoperimiric polygons having the same number of sides, the maximum is equilateral. 232. Theorem IV. Of all polygons with sides all given but one, the maximum (in area) can be inscribed in a semicircle having the undetermined side for its diameter. 233. Theorem V. Of all polygons with the same given sides, that which can be inscribed in a circle is the marimum. 234. Corollary 1. Of all isoperimetric polygons of a given number of sides, the maximum is regular. 235. Theorem VI. Of to isoperimetrir regular polygons, the one having the greater number of sides has the greater area. 236. Corollary 1. The circle is the maximum of all isoperimetric plane closed figures. 237. Theorem VII. Of all regular polygons of the same area, that which has the greatest number of sides hus the minimuin perimeter. 238. Corollary 1. Of all plane closed figures that rere equal in area, the circle has the minimum perimeter.

·INDEX [Numbers refer always to pages.) Altitude, of prism, 239; of pyramid, 280; lateral area of, 265; volume 253; of frustum of pyramid, 254 ; of, 266. of cylinder, 263 ; of cone, 268; of Cylindrical surface, 263. zone, 311. Angle, convex polyhedral, 232; di- Diagonal, of polyhedron, 238; of hedral, 228; of spherical polyspherical polygon, 297. gon, 297; plane, 228; polyhedral, Diameter, of sphere, 281. 232; spherical, 296; symmetric Dihedral angle, 228; measure of, trihedral, 308; trihedral, 232; 228; plane angle of, 228; equal, units of, 202. 228; faces of, 228; edge of, 228. Area, lateral, sce Lateral aroa; of Dimensions, Geometry of three, sphere, 310; of zone, 311; of 215; of a rectangular parallelelunc, 312, 313; of spherical piped, 248. triangle, 317. Directrix, of cylindrical surface, 263. Axis, of cone, 268; of circle of a Distance on a sphere, 288. sphere, 288. Dodecahedron, 238. Edge, of dihedral angle, 228; of Base, of prism, 238; of cylinder, polyhedral angle, 232; of poly- 263. hedron, 238. Birectangular triangle, 304. Element of cylindrical surface, 263. Equivalent solids, 245. Cavalieri's theorem, 2ts. Filce, of dihedral angle, 228; of Center, of sphere, 281. polyhedral angle, 232; of poly- Circumscribed sphere, 294. Cone, 268; altitude of, 268; axis of, Fice angle, 232. hedron, 238. 268; circular, 268; right circular, Frustum, of pyramid, 254; of cone, 269; slant height of, 269; frus- 269; lateral area of, 255, 271; tum of, 269; lateral surface of, volume of, 261, 262, 272. 269; of revolution, 269; prismoid formula for, 280; volume of, Generatrix, of cylindrical surface, 272. 263. Congruent polyhedral angles, 232. Great circle of spherc, 288. Congruent solids, 242. Conical surface, 268. Hexahedron, 238. Cube, 245; volume of, 249. Cylinder, 263; circular, 263; right, Icosahedron, 238. 263; right circular, 263; of revo- Inscribed sphere, 294. lution, 264; prismoid formula for, Isosceles spherical triangle, 297. xlvii ·xlviii INDEX Lateral area, of prism, 238, 241 ; of | Projecting plane, 237. regular pyramid, 255; of frustum Projection, 237. of regular pyramid, 255; of any Pyramid, 253; lateral faces of, 253; cylinder, 265; of right circular vertex of, 253; altitude of, 253; cylinder, 265; of any circular regular, 253; slant height of, 253; cylinder, 205; of right circular truncated, 254; frustum of, 254; cone, 271; of frustum of right inscribed, 270; prismoid formula circular cone, 271. for, 280, lateral area of, 255; Lateral face, of prism, 238; of pyra volume of, 261. mid, 253; of frustum of regular pyranid, 254. Quadrant, 291. Lune, 312; arca of, 312, 313. Radius, of a sphere, 284. Oblique prism, 239. Regular polyhedron, 276. Octahedron, 238. Regular prism, 239. Regular pyramil, 253. Parallel lines, 219. Regular solids, 276. Parallel planes, 219. Right prism, 239. Parallelepiped, 245; right, 245; Right section, of prism, 239; of rectangular, 245; oblique, 245; cylinder, 263. volume of, 250. Perpendicular planes, 226. Similar polyhedrons, 276. Plane, 215; how determined, 215; Similar tetrahedrons, 275. line perpendicular to, 219; per- Slant height, of pyramid, 253; of pendicular planes, 226; parallel frustumi of regular pyramid, 254 ; planes, 219; projecting, 237. of cone, 269; of frustum of regular Point of tangency, 285. ('one, 269. Polar distance, 291. Small circle of sphere, 288. Polar triangle, 300. Solid angle, 317; measure of, 317. Pole of circle of sphere, 288. Solids, congruent, 242; equivalent, Polygon, spherical, 297. 245 ; regular, 276. Polyhedral angle, 232; convex, 232; Sphere, 284 ; center of, 284; radius congruent, 232; faces of, 232; of, 284 ; diameter of, 284; tangent edges of, 232; vertex of, 232. plane to, 285; tangent line to, Polyhedron, 238; edges of, 238; ver 285; great circle of, 288; small tices of, 238; faces of, 238; diag circle of, 288; distance on, 288; onal of, 238; similar, 276; regular, inscribed, 294 ; circumscribed, 294 ; 276. area of, 310; volume of, 318. Postulate, on the plane, 216; on the Spherical angle, 296. cylinder, 264. Spherical degree, 315. Prism, 238; bases of, 238; lateral Spherical excess, 315. faces of, 238; lateral area of, Spherical polygon, 297. 238; altitude of, 239; right, 239; Spherical triangle, 297 ; area of, 317. oblique, 239; regular, 239; right Symmetric triangles, 305. section of, 239; congruent, 242; Symmetric trihedral angles, 308. truncated, 243; volume of, 251; inscribed, 264; prismoid formula Tables, i-xxvii. for, 280; lateral area of, 241. Tangent line to sphere, 285. Prismoid, 281. Tangent plane, 285. Prismoid formula, 280. Tetrahedron, 238; similar, 275.

·INDEX xlix Triangle, birectangular, 304; spheri- Volume, of rectangular parallelecal, 297; symmetric, 305 ; tri piped, 248; of any parallelepiped, rectangular, 304. 248; of triangular prism, 251; of Trihedral angle, 232; congruent, 232; uny prism, 251; of triangular measure of, 315. pyramid, 260; of any pyramid, Trirectangular triangle, 304, 261 ; of frustum of pyramid, 261, Truncated prism, 243. 262; of any cylinder, 266 ; of Truncated pyramid, 25-4. circular cylinder, 266; of any cone, 272 ; of frustum of cone, 272; of Vertex, of polyhedral angle, 232; sphere, 318. of pyramid, 253 ; of spherical poly- Zone, of sphere, 311 ; arca of, 311. gon, 297. Printed in the United States of America.

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