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·SOLUTIONS TO PROBLE
CONTAINED IX
A GEOMETRICAL, TREATISE
ON SONIC SECTIONS.
BY TUK
REV. 1. Il. DREW, M.A.
OF ST. JOANH COLI.FCE, CAMBRIDGE ;
BOXIX MASTER OF BLACKHEATA PROPRIETARY SOHOX
Cambridge:
MACMILLAN IN D CO.
AND 23, HENRIETTA STREET, COVENT GARDEN,
mon.
1882

·Liturpere krimet
Auch. No 21714. Dan
4 le
k LA Y, , ANH TAYLR, PINTins, LUVI20

·P'REFACE. more Tur present work is designed to be a continuation of, and is intended to be used by the learner in connexion with, my treatise on “Geometrical Conic Sections." The object is not so much to furnish a key, as to illustrate the application of the principles of Geometry to the solution of questions in Conic Sections. It has been too much the custom to despise the use of geometrical methods, and to reject them as uncertain except in the solution of very clementary problems. The present work will, I hope, show that Geometry is a effective and practical instrument in the treatment of this subject than is generally supposed. Another object that I have likcwise kept in view is to supply a comecting link by which the student may be led from the simple process of following the steps of a demonstration, to devising for himself the solution of an origmal geometrical problem. Most persons engaged in teaching must have felt the want of some intervening step to bridge over this difficulty. With this purpose the Solutions are all based on one or other of the propositious in the former trcatise, and the student is left to ·PREFICE. modify the fonstruétion and figure dccording to othe particular question under consideration. An amount of care and attention will thus be rendered pecessary, which cannot fail to įnsure the complcte understanding of the principles of the subject, and which will make .it impossible for the learner to acquire 'an apparent but treacherous knowledge, by the updzie use of a retentive memory. Prob. 31 of the Parabola, Prob. 55 of the Ellipse, and Prob. 32 of the Hyperbola, afford good illustrations of the plan I have pursued, which really indicates the way in which the properties were themselves suggested. / It has been found convenient to refer occasionally to a new edition of the “Conic Sections," whiclris now in course of publication. In this edition such slight modifications, as experience has shown to be needed, will be introduced. A considerable number of unsolved problems will be added, and the work will be brought completely up to the requirements of the present time. I am indebted for much valuable help in the preparation of these “Solutions,” oto Mr. A. Freeman, of St. John's College, and to Mr. F. R: Drew, of Sidney College. W. H. DREW. BLACKHEATH PROPRIETARY SCHOOL, Vorember 200h, 1861.

·CONIC
SECTIONS
1859
PROBLEMS ON THE PARABOLA.
1. DRAW BQ at right angles to AB; then
AS. SQ = BS = 4 AS",
:: SQ = 4 AS, and A Q = 5 A $. ,
2. The triangles PNG, GPK are equiangular, and have
the side PG common,
.. they are equal in all respects,
VG 2 AS.
.. PK
3. SP- SG = 2 NG = 4 A8= BC.
4. I'roduce PQ to meet the axis in T, and join A Q; then,
ST. SA = 8 Q4 = 4 AS,
... ST= 4 S, and S.V= 2 AS,
::S:SQ :: SV: SP,
.: LSQA = L SPN, or LOSB = L PSB.
5. Since PSZ is a right angle, and S Y perpendicular to PZ,
.:. PY, PZ= SP,
also PY. YZ= SY = SA.SP.
6.
AN.NL = PN = 4 AS. AN,
... NL=4 AS.
7. Let the tangent at P meet the latus rectum irr L; then
ZY: YL: XA: AS,
B

·CONIC SECTIONS. ŻY = YL, and S Y is perpendicular to ZL, ..:: SZ= SL. 8. Since AT = AŅ, and the Angles at A are right angles, ... TY= NY. Also, since the angles atºš and Y are right angles, a circle may be described about SNPY, ... FP', TY = TN. TE 9. Let the tangent at P meet the tangent at A in Y; then since SYP is a right angle, the circle described about SNP will pass through ř. Also, since Al" = AS. AT= AS. AN :: A Y'a tangent to the circle. Now AZ: PN :: AT: TN, .:. AⓇY = PN. 10. Let the tangent, ordinate, and normal at P meet the exis inf T", N' and G'; then since AN = AT" and N'N= NG", :: AN= } T" G'. Now PN = 2 AS. 2 AN= A" G'. E'G' = P' G", .:. PNP' G'. 11. From any point R on the tangent at P draw the tangent RQ; then the triangles SQR, ŠPR are similar, ... the angle SRQ the angle SPR, .. the SRQ is independent of the point R. 12. Draw NQ a tangent to the circle. Now RN : AB :: AN : A A", and PN? : A' Bº :: AN: A A'; :: AN. AA : A A", .:. RN - PN : AB : AN. AN: A A", or RP. RP : NQ :: AB : A A'?.

·PROBLEMS. 3 13: Let B he the point where the axis of thê parabol& is to touch the circle, and the other given point. Draw the tangents TC, TBG intersecting in T'; also draw EN perpendicular to TG and CG passing through the centre O of the circle; then NT and NG will be respeetivdy the subtangeht and subnormal.of the parabola required. Bisect NT in and TG in S; then A and S will be respectively the vertex and focus of the parabola. .. 14. Let QT be the tangent at the point Q. Througe Q draw the chord QOQ'; and let WRO be drawn parallel to the axis meeting QT in W, the parabola in R, and the chord Q Q in 0. Bisect Q Q in V, and draw the diameter TPV parallel to the axis, and join SP. Now from similar triangles QO: OIV :: Ql': V1, :: Q1" : QV.I'T, :: 4 SP. PV : 2 QV.PV, since Q1* = 4 SP. PV (Prop. XV.) and VT= 2 PV (Prop. XIV.) .. QO: Ow:: 4 SP: 2 QV, :: 4 SP: QQ, I 20.QQ = 4 SP.. O W. But Q0.0Q = 4 SP.RO (Prop. XVII.) ... QQ:OQ :: OW: RO, .. QO: O Q :: WR : RO. 15. Join SP; then PV = ST= SP= PO. B 2 ·4 CONIC SECTIONS. 16. Draw VM at right angles to the axis ; then šince VM: MS :: PN': NT, ... MS = NT. But PN = 2 AS. NT, VM 21s. SM, .. the locus of V is a parabola, whose vertex is S, and latus rectum half that of the given parabola. 17. Produce Q1', UK to meet the tangent at Pin Il' and X; then as in Prop. XIX. it can be prove that IIT and XU are each equal to 4 SP, .WV and XU are equal and parallel, 9. UV is parallel to the tangent at P. 18. By similar triangles CH: CA:: AV: PN, .:: PN, CR 10. AV, or P.V = 10. A.V, :. the locus of P is a parabola whose axis is 1 B, and latus rectum equal to the given distance - C. 19. Let CP and AR intersect in 1'; and draw INI to A C, and I'M I to the tangent at A ; then VN: AN:: CR: CA, :: CR : CP, :: VN: CV, cy= AN. = AN= VM, the locus of V is a parabola whose focus is C, and directrix AM. 20. Let OQ, O Q be two equal tangents drawn from the point 0 in the axis; and let them be cut by the tangent RPR in R and R'.

·PROBLEMS. 5 Join SP; SQ, SQ'; SR, SR'; then the L SOR' = the L 80Q = the L sQ0; and the L SR,0 = the supplement of SR' Q', SPR' (Prop. XII.) the-angle, SPR, the angl. SRQ (Prop. XII.) .. in the triangles SOR', SQR, the angles®SOR", SRO) = SQR, SRQ each to each, and S() = SQ, ..RO and OQ = 0 Q', ... OR = R' 0'. OR', 2 21. Join QE, QE; then the LQEQ = 2 L QOQ'. Agaill let 0Q and OQ meet the axis A Sx in T and l"; then LQSC SQT + STQ, = 2 L STQ = 2 L QOV; so the Ļ Q! Sx = 2 L ST"Q = 2 L QOV, ... the LOS 2 L QO0, .. the LQ8Q = the LE', ... the circle described about the triangle QEQ will pass Irough S. If Q & pass on the left of S, a similar proof will hold ood. It must then be shown that the angles QSQ, QEQ are ogether equal to two right angles. 22. Draw the ordinates PN, pn; then DN: On :'Or: sp, :: XN: Xn, ·CONIC SECTIONS. Sn, .. SN : XN :: Sn : Xn, ..XN +SN: XN - SN :: Xn + Sn : Xn .: 2 AN: 2A8 :: 2 AS: 2 An, or AN : AS :: AS: An, .. 4 AS. AN: 4 AS :: 4 A8 : 4 AS. An; • or PN : 4 AS ::4 AS : pn, ... PN : 2 AS :: 2 AS: pn, or PN: 4 AS :: AS: pn. Again, SQ:AS:: PN : AN, :: 4 AS. PN: 4 AS. AN, :: 4 AS. PN: PE, :: 4 AS: PN, :: pn : AS, :: SQ = pn, 80 Sq = PN. 23. Join QS and produce it to q; then SR bisects the angle PS4, }(Prop. IV.) • . and Sr :: the angle R Sr is half of the two P Sq and p Sq, .. the angle R Sr is a right angle, ... DR. Dr = SDP = 4 AS. 24. Join PQ, and draw 0 V parallel to the axis; then since OQ : 0Q :: PV: QV,. .. Q'Pis parallel to OV, f : the LOQ P= the LQ0V, = the L QT8, if o meet the axis in T, = 'the L SQO, and the LOPQ = the L OPS, = the L 80Q (Prop. XII.)

·PROBLEMS. :: the triangle PO Q' is similar to OⓇQ, :: PQP: 0P :: 00:08, . 08. PQ OQ.OP. 25. Draw Q V parallel to the tangent at P; and draw SY perpendicular to the tangent, and join A Y'; then QD: QV :: 48Y: ST, :: SY : SP, : 00 : QV :: SY : SP, :: SP. SA: SP, :: SA: SP, :: 4 AS: 4 SP, :: 4 AS. PV : 4.SP.PV, .. QD= 4 AS. PV. 26. Draw the tangent PT at the point P meeting the axis in T'; then by Prob. 14 TA:AO :: PO: O Q. But AT= AM .. AM: AO:: PO:OQ; So AN: 10:: OQ : OP, .. AM: AI :: AO : AN, ... AM. AN= A O. 27. By similar triangles AM: PM :: QN: AN, AM' : PM' :: QN : AN, or AM: 4 AS. AM :: 4 AS. AN: AN, or AM: 4 AS :: 4 AS: AN, .: AM. AN= (4 A 8).. 28. Let Q be the centre of the circle inscribed in the sector OAP ·8 CONIC SECTIONS. Rraw 0 9R, meeting the circles at their point of.contact R. Complete the quadrart A OB, and draw BN parallel to 0A; and through draw MQN, meeting 0 A and BN at right angles in M and N; then QR = QM, and OR = MN, OQ= ON, .. the locus of Q is a parabola, whose focus is (, and directrix BN. 29. (1.) AN: AM :: PN : QM? :: AN : SM; or AN : AN. AM :: AN: SN, .:: SM';= AN. AM. (2.) Bisect QQ in V, and draw VR parallel to the axis bisecting AP in R; then AS + AM = SQ, AS + AN=SQ', .. MM SQ - SQ', 2 SV = 2 AR, AP. . 30. RV: QV :: PV : PV', :: QV: V', RV.QV : QV' :: QV : Q'V', ... RV. QV = QV; So R'V'. QV = 'V', .. RV : 'R'V' :: QV : Q V's. 31. In the fig. of Prop. XVII. let Qq be the diameter of the circle, and Q'q.the chord joining the two other points of intersection of the circle and par&bola. c ·PROBLEMS. g Jon VV' meeting the axis in E: V'V' is at right angles to Q! Now since the rectangle Q0.09 = 90. Od, {.. PV and P V are equally distant from the axis, ::. VE = L'E. Let 24. & meet the axis in F and F"; and draw VM, VM at right angles to the axis; then FF MM' = 2 EM = 2 subnormal of P=4 A$. w 32. Draw OPV parallel to the axis bisecting QQ in V; then since the tangent at P is parallel to QQ, it will be at right angles to ( Q, and will therefore meet OQ on the directrix. Let this point be Z; then OZ: ZQ :: OP: PV, .. OZ=ZQ. 33. The parabolas will intersect at the extremities B, C, of the latus rectum. Draw the normal BG; then the angle.at which the parabolas intersect is twice the angle SBG, which is a right angle, since SG = 2 AS = SB. 34. Let BSB' be the chord of the circle of curvature at B through the focus. Bisect B in C, and draw CO at right angles to BB”, mecting the diameter of curvature in 0; then since CO is parallel to the axis, BO: BG :: BC: BS and BC= 2 BS, BO = 2 BG. 35. Draw PW parallel to the axis'; then the angle SPF= WPH = SHP, and the angle at S is common to the triangles SPF, SHP; .. these triangles are similar, .. SF: SP SP: SH, .. SF. SH-SP = SG*.

·10 CONIC SECTIONS. 86. Let P Q be the inscribed triangle. Draw the tangents Qo, QO, meeting in 0; and the tangent tPť meeting 00, O Q in t and to Produce QP, QP to meet a Q, O Q' in q and q. Join o'q; then, what we have to prove is that qq', and the Tangent at P will meet @ produced in the same point. Draw OV, PM, qm, qm, tn, t'n', parallel to the axis of the pårabola, meeting og in V, M, m, m, n, n'. Produce MP to meet O Q in R, and let qm, q'm' meet tPť in r and r'. The problem will clearly be solved if we can show that qr : qm :: g'n' : qm'. Now qm : Qm :: RM : QM, Q'm : qm :: QM: PM, Q'm : Qm :: RM.QM : QM.PM; but RM : PM :: QQ: Q'M (Prob. 14.) ... Q'm : Qm :: QQ.QM: QM.QM, :: QQ: QM, .: Om : QQ:: QM: QQ + QM. Again, : Pt, :: 2mn : 2 Mn, :: 2 m 2 On : 2 Mn, :: 2 Qm - QM : QM (since tn bisects QM); but 2 Qm : QM :: 2QQ: QQ + QM, .:. 2 Qm - QM:QM:: QQ QM : QQ + QM, .:.qr : PR :: QQ- QM: QQ+ QM, and PR : PM :: QM: Q'M, ..qr : PM :: QM : QQ + QM, :: Qm: QQ, .. qr : Qm :: PM : QQ. In the same manner it can be shown that op gel : Qm' ::. PM : QO', .: qr : q' ::'Qm : Qm. qr : PR :: 5 ·PRODLEMS. 11 Bot gm :: Qm : QV: OV, :: Qm' : qm', .. Que : Q m' :: qmº: qm', gr: qqm : q'm. or qr : qm :: a'zo : a'm'.• : 37. Let the tangent RR meet the curve in P; "and draw Km, PM, R' m' parallel to the axis meeting Qin m, M and m'. Now 'OR: RQ:: Vm : Qm, :: 2 Vm: 2 Qm, :: 2 QV - 2 Qm : 2 Qm, :: QQ-QM: OM, :: QM: QM. Similarly OR : RQ :: QM : QM, ..OR: RQ:: Re: OR'. 38. Let AQ and AQ be the chords, and R the further angle of the rectangle. Join QQ, and draw AR bisecting it in V. Also draw VP parallel to the axis, and at P draw the tangent PT parallel to QQ. Draw PN, VM, RL at right angles to the axis. Make A O' = 2 AO = 8AS (Probs. 26 and 27); Then RL = 2 VM = 2 PN, and O'L - AL - AO = 2 AM - 2 A 0, = 2 OM = 2 TN='4 AN. But PN = 4 AS. AN, ... 4 PN = 4 AS. 4 AN. or RL' = 4 AS. OL, the locus of R is a parabola equal to the original one with its vertex at O'.

·12 CONIC SECTIONS. 39. Draw A H at right angles to the tangent at P; then if PK be the chord of curvature through A, PR : PU ::.All : AP (Sce, fig. Prop. XIX.) ... PK. APA PU, AH. Again, PU :4SP :: 'SP: $(Prop. XIX.) :: 'TY: AY, .. PU: PY :: 4 SP: AY (since PY TY). Also, AH: PY :: AY: SP, :: PU. AH : P1" :: 4SP. AY » SP. AY, .:: PU. AH = 4 PY". .. PK. AP=4PY. O 40. This problem is incorrect. 41. Let Q Q'. be a chord. Bisect it in V, and draw QM, VK, QM' at right angles to the directrix; then 2 VK= QM + OM = SQ + SQ. But SQ + SQ > QQ, unless Q Q pass through S, .: VK > QV. Hence the circle described upon QQ as diameter will only reach the directrix when the chord Q Q passes through the foeus, in which case it will touch the directrix, since the angle VKM is a right angle. C 42. Let A and A' be the vertices of the parabolas. Produce the axes to meet in T, and draw the common tangent TP. Join 88', cutting PT at right angles in Y. Join A Y, and draw S'X perpendicular to ST.

·PROBLEMS
13
Now since A Y is perpendicular to sx,
.:: SY : 'S'Y :: SA: A.X.
But
SY = SY,
... SA – AX,
:: S' is on the difectrix.
43. In the fig. of Prop. XIX. let be the other point
where the circle meets the parabola, and draw Q X'R' parallel
to the axis, meeting the circle in X' and the tangent at P in
R'; then
XX' is parallel to RPR (Prob. 17)
and when the circle becomes the circle of curvature,
WX' is parallel to RPR",
.. WX' is at right angles to PU.
Now if the points Q and U coincide,
IŅU and P W will be equal,
and WX will bisect PU,
and PWUX' will be a square,
... the L GPS= the L GPW,
the L GPN,
... Pis at the extremity of the latus rectum.

·14 PROBLEMSON THE ELLIPSE. al. Draw the diameter D parallel to the tangent at P, meeting SP in E. Also draw the normal PGF perpendicular to CD, and meeting it in F. Then the angle sP'S' = 2 the angle S'PG. Now since PE is always equal to AC, it is evident that the angle SPG will be greatest when PF is least. But PF. CD= AC.BC (Prop. XXII. Cor.) .. PF is feast when CD is greatest, that is, when CD=AC and P is at the extremity B of the minor axis. 2. Draw the latus rectum LSL'; then AC: : BC :: AS. A'S : SL (Prop. XIII.) or ACS : BC? :: AchN? : SL (Prop. IV.) or AC : BC :: BC: SL", : BC :: BC : SL, .. AA : BB :: BB : LL', .. L L' is a 3d proportional to A A' and BB. or AC 3. Draw BH parallel to LS, meeting the major axis in H; then ABCH: ALSS :: BO : SL* :: AC: BC”. (Prob. 2.) Take CK = CH; then rectangle BB'. CK = A BCH, ... BB. OK: A LSS :: AA : BB'*. 4. Let the tangent at to one of the ellipses meet the minor axis produced in t; then ·PROBLEMS. 15 Ct. SL = BC (Prop. XIV.) ... Ct :'BC :: BC: SL :: AO: BC (Prob. 2.) ... Ct•£ AC. Hence Ct is constant, and the tangents will all pass through t. 5. In this problem the ellipses are to have their major axes equal as well as a common focus. Let S be the given 'focus, and Y PY' the given fixed line (see fig. Prop: XV.); then since S and YPY' are both given in position, the point Y of the perpendicular SY will be a fixed point. Now YC = A C (Prop. XV.) Hence the locus of C is a circle whose centre is Y and radius 10. 6. Draw SY, S V" perpendicular to the given line (see fig: Prop. XV.), and take BC a mean proportional between SY and S Y'; then BC is the semi-minor axis. (Prop. XV.) Also, since A C'* = B C + CS", AC is also known; and since Cx : CA :: CA: CS (Prop. II.) the position of the directrix is known, and the ellipse may be described. 7. See fig. Prop. IX. TQ : PN :: AT : AN, TO: PN :: AT: AN, Again, CT. CN.= CAP, .. CT: CA':: CA: CN ·16 CONIO SECTIONS. : CT+CA: CT-CA :: CA + CN : CA - CN, or A'T: AT :: A'N : AN, ..A'T: AN:: AT : AN, Hence TQ : PN*:: TO: PN, TQ = TQ. 8. See fig. Prop. IV. If SBS' be a right angle, SC and CB are equal, .. ACP = SB? SC + BC2 2 BC". > 9. Draw the ordinates QM, PN; then QM: AM :: PN : CN, ... AM. AM : AM° :: A C2 CN? : CN or A'M : AM :: AC? CN : CN, .. 2 AC : AM :: AC : CN, .:. 2 AO: AQ :: AO: CP, or 2 A02 : A0, AQ :: A 0% : CP, :: AO. AQ = 2 CP c 10. If AB and CD are equally inclined to the axis, the diameters parallel to them will also be so, and will be therefore equal. Hence if AB and CD or these lines produced meet in o, OA.OB = 0C.OD. Hence a circle may be described about A, B, C, D; and if AC and BD or these lines produced intersect în O', 04.00= OB.OD, .. AC and BD are equally inclined to the axis. So with regard to A-D and BC. "11. Draw PM, pm at right angles to the directrix. Produce MP, QS to meet in, and mp, qS to meet in q'; then ·17 PROBLEMS. QP: PM :: $4.: : AX, · QP. SP, .:: SQ bisects the angle P8A'; So sa PSA, .. Q Sq' is a right angle. .. also. Q8q is a right angle. 12. Draw the diameter SK cutting the tangent at B at right angles in 0; then SK. SO = SB, or SK. BC= AC?. 13. Through Q draw RQR' touching the inner circle in Q: and cutting the outer in R and R'; then PQ.P'Q QR" = CR CQ, = AC- BC = CS". 14. Draw the ordinate PN, and the semi latus rectum SL then by similar triangles SL: SN :: SG : SP, :: SA: AX; Also SP: XN. SA: AX, .. PL: SX :: SA: AX; But SL : SX :: SA: AX, ... PL SL'. 15. Draw the normals PG, DG"; then PG" : CD' :: B C : AC? (Prop. XXIV.) and D G : CP :: BC : AC, ... PGP + DG" : CP + CD :: BO? : AC'. But CP+CD= AC + B C*** :: PG* + D G" : A C + B 09 :: BOY : AG ... PGP + D G2 is constant.

·18 OCONIC SWCTIONS. 16 Produce PG to meet CD in F, and join CP; then CQ*°= CP® + PQ – 2 PÉ. PO (Euclid, II. 13) CP + 2 CD-2 PF, CD, A CS + B C – 2 AC. BC (Props. XX. XXII.) F. CQ=AC : BC. 17. Let the tangents at Pand P intersect at right angles in 0. Draw CK, CK' at right angles respectively to OP and OP'; then COM CK + CK". Again, from S draw S Y and SY' perpendicular respectively to OP and OP'; and join CY,*cº". Also let SY' and CK intersect in Il; then CK + SH' = CY = AC, and CK" + CH = CY" = A C", ... CK + CK + C$* = 2 AC, .. CK + CK = A C" + BC", .. CO* = A + B O*. 18. Draw the ordinate QM; then NR : AN:: QU: AM, and NS: AN :: QM : AM, .:: NR. NS: AN. AN:: QM : AM. A'M. But PN : AN. AN:: QM: AM.AM ... PN = NR.NS. 19. By similar triangles GL: PN :: SG : SP, :: CS : CA (Prop. XI.) 20. Since the tangent'at L meets the directrix in X, QN: XN:: SL: SX, :: SA: AX; Also SP: XN:!, SA: AX, .. QN = SP.

·PROBLEMS. 19 21. See fig. Prop. VII. PZ bisects the L SPW, :: the L SPZ is the L MPZ. 22. See fig. Prop. XXIV. Let S'P meet CD in E. Produce PF to meet the axis minor in K, and join EK; then PK: PG :: NC :NG, :: AC: BC" (Prop. XII.) : 'PK. PÅ : PF. PG :: A 0 : BC", .:: PK. PF = AC" (Prop. XXIV.) = PE (Prop. XV. Car.) .:: the L PEK is a right angle. So also if P S produced meet CD' in E", and E' K be joined, the angle PE' K may be proved to be a right angle. .. the perpendiculars EK and E' K meet the minor axis in the same point. 23. N G : NC :: BJ : A Ở* (Prop. XII) .: CT.NG: CT.CN :: BO: AC :: OT.NG = BC (Prop. IX.) ... CT : BC :: BJ : NG. .. 24. A circle may be described about PYSN; .:: LPNY = L PSY. So L PNY = L PSY', and L PSY = L PS' Y", .: LPNY= L PNY", .. PY: PY' :: NY: NY. 25. See fig. Prop. XV. Draw CZ at right angles, to the tangent meeting S'P in K; then by similar triangles C 2 ·20 KONIC'SESTIONS. KE : CE :: SP: PY. But CE=PY, .: KE=SP. 26. Let R be the point where the circle described about QPQ meets the ellipse; jdin PR meeting Q Q' in 0, and draw the diàmeter CP'.parallel to PR, and CD parallel to QQ'; then from the ellipse 20.00 : PO.OK :: CD :CP" (Prop. XXIX.) and from the circle Q0.0 Q PO. OK, .: CP= CD, CP being fixed in position, PR, which is drawn parallel to CP', meets the ellipse in a fixed point. 27. Produce QP',Q'P' to meet the tangent at Pin R and R'. Draw the diameter CD parallel to the tangent at P; also draw the diameters CE, CE parallel respectively to QP and Q'P'; then it is evident from Prop. XXIX. that RP.RQ : RP :: CE : CD, w R'PR'D': IR'P :: CE*: CDa. But CE', Since P Q and P Q are equally inclined to the axis, .. RP.RQ : R'P', R'Q' :: RP2 : R'P2 But RP: R'P :: RP : RP, since P'P bisects the LQP'R, .:: RP.RQ: R'P'R'Q' :: RP'': RP .. RiQ : R Q :: RP : RP, .: QQ. is parallel to RR'. 28. See fig. Prop. XXII. Let a parallelogram LPll'l be formed by drawing tangent at the extremities of a pair of cônjugate diameters PCP', and DCD', CE = ·PROBLEMS. 21 On the portion of the ellipse DPP', take any point Qe and at the extremities of the diameter Q0Q', draw the tangents RQr, R Q'r' meeting the tangents at D and D' in R, R™ and Ti m' respectively. 'Also let Řr intersect Li in O. Now, since Ll is bisected in P, it is evident, that on whichever side of P the point is taken, the parallelogram RT is greater than the parallelogram LL', and that the excess is twice the difference of the unequal triangles R01, r OL. Hence the parallelogram formed by drawing tangents at, the extremities of conjugate diameters, (which is, of constant area) is the least which can be described about an ellipse. 29. (CP+ CD) = CP' + CD + 2 CP. CD, ACP + BC + 2CP. CD (Prop. XX.) Hence (CP+ CD)' is least when CP. CR is least. Now CP.C.D.is > PF. CD, unless PF = CP, and PF. CD= AC. BO"(Prop. XXII. Cor.) Hence CP. CD is > AC.BC, unless PF= CP; and when PF = CP, CP. CD= AC. BC; :. CP. CD is least, when PF and CP are equal, or when CP and CD coincide with the axes of the ellipse. more 30. Let S be the given focug, and P the given point. Join SP, and on SP produced make SW equal to the major axis AA. With centre Pand radius PW, describe a circle W U. The other focus of the ellipse must be on this circle. Now, since in an ellipse CS = A012 BO* (Prop. IV.) .. SS'? BB'. Hence we have the following construction for finding & the other focus and centre. On SW as diameter describe a semicircle, and in it place WH equal to the minor axis BB'. = AA' ·22 CONIC SR_TIONS. With centre S and radius SH describe a circle cutting the circle WU in.S'. Then SS'S = SH SW WH = AA" – BB”. .. S'ia the other focus. Bisedt SS' in C; then C is the centre. 31. Let Q be any point on the ellipse, and let QP produced meet CD in M, and let QP' meet CD in N. Also draw QV parallel to LD, and QR parallel to CP. Then CM: CP :: QV: PV, CN: CP :: QV: PV, :: CM.CN: CP :: Q1":PV.PV, :: CD: CP2 (Prop. XXI.) ...CM. CN= CD. .. :: AC : 32. Let CP and CD be conjugate diameters. Then SP: XN :: SA: AX (See fig. Prop. IX.) :: CA: CX (Prop. II.) .: SP: AC :: XN: CX, .: AC - SP: AC :: GN: CX, or AC SP CN СХ. So also if DN be the ordinate of D, AC-SD: CN' :: AC: CX, ..(AC-SP) + (AC-SD)': CN' + CN". CA: CX*, :: C8 : CA'; But CN + CN'' = CA* (Prop. XX.) :: (AC - 8P)* + (AC – SD)* = CS?. : 33. Draw the ordinates PN, DR, and produce them to meet the auxiliary circle in PSD ; also let CB produced meet the circle in B'.

·23 PROBLEMS. Join B'D', B'P'; A'P', AD, and let A'P', AD intersect in O'. Now, since P'D' is a quadrant of a circle, the arc AP6= the arc B'D', :: B'Pºis parallel to AD:; So also B'D' is parallel to A'P', .: B'O' is a parallelogram. And since BP and B'P' produced meet the axis in the same point, and B'P' is parallel to AD', and BC: B'C:: DR: D'R, .. BP is parallel to AD; So also BD is parallel to A'P, :: OB is a parallelogram. Again, join PD, P'D. These lines will, when produced, meet the axis in the same point. Now if any straight line be drawn at right angles to A A, meeting PD, PD, it is easily seen that the portions of this line intercepted by the triangles PBD, P'B'D' respectively are in the proportion of BC to AC. Hence the triangles PBD, P'B'D' are also in this proportion. .:: ODBP: O'LB'P :: BC: AC. Now since P'D' is a quadrant, the triangle P'B'D will have its greatest value when B' is half way between P' and D, for the perpendicular from B' on P'D' will then be greatest. Hence ODBP is greatest when CP and CD are equal. 34. PS. Sp: AS. AS:: CQP: AC (Prop. XXIX.) .: PS. Sp: CQ. Cq :: AS. AS: AC", CS2: 10, :: BO: A0%. :: CA 35. See fig. Prop. XIX.

·24 PONIC SECTIONS. ÅT: AC :: PN: CN, :: CR:'DR (Prop. XIX. Cor.) :: AC: At, ..AT.AtE AC. 36. Draw the diameter co, cq, Cr parallel to the tangents ; then PR : PQPT: Cyl: CQ (Prop. XXIX.) or P'R : P'Q :: Cr : Cq. So QP:QR :: Cp : Cr, and R'Q: R'P :: Cq : Cp, .. P'R, QP. R'Q = P'Q. Q'R. R'P. 37. See fig. «Prop. IX. Let the tangent at P meet the tangent at A in Q; then QP and QA subtend equal angles at S (Prop. XVII.) : QS bisects the L TSP, and PT bisects the L SPW, .: Q is the centre of the circle ST, SP and S'P produced, the locus required is the tangent at the vertex A. 38. It is easily seen that this question is only the converse of Prob. 33. * 39. Let A CA', a Ca' be the major axes of the ellipses, inclined to one another at any angle. Let P be the point where the ellipses intersect within the angle A Ca, and the point within the angle A'Ca; then since CP is a common semi-diameter of both ellipses, .. CP makes equal angles with CA and Ca, CP bisects the angle A Ca, So CQ bisects the angle A' Ca, :: the LPCQ is half of A Ca and A' Ca, .: PCQ is a right angle.

·PROBLEMS. 25 QN ..then 40. Draw the ordinates PM, PM: QŇ :: SP: SQ, :: XM: XN, .: PM: XM :: QN: XN, .. the LPXM - the L QXN. .. 41. Dravý the ordinates":PM, QN; and let Om ca be the diameters parallel to the tangents at P and Q respectively ;. then RP.RP.: RQ. R'Q :: XM. X'M: XN. X'N. Now XM:SP :: CA:CS:: XN:SQ, and X'M : S'P :: CA: CS:: X'N: S'Q, .: XM. X'M: SP. SP :: XN. X'N: SR. S'Q, .: RP.RP:RQ. R'Q :: SP.SP: 8Q. S'Q, :: Cp* : Cq* (Prop. XXIV.) :: OP: OQ (Prop. XXIX). 42. See fig. Prop. XIV. Let the normal to the ellipse at P meet CQ produced in R'; then the triangles R'PR and TP.Q are similar. .:: H'R: TQ :: PR: QP, :: CQ: TQ .:: R'R = CQ= AC, .: CR' = CR + AC, = BC + AC (Prop. XIV.) .. the locus of R' is a circle, whose centre is C and radius AC + BC. 43. See fig. Prop. XV. Let CD meet SP produced in E, and S'Pin E ; then S'E' = S'P-PE = S'P-AC, and S E = PE - SP = AC - SP.

·26 LCONIC SECTIONS Byt S'P + SP = 2 AC, - SP-AG = AC - SP, :: SE= S'E'. Again, if D and D be the diameters of the circles described about SCE and SCE, since the perpendiculars from S and S' upon CR are equal, .. D:D :: WS. SE: CS. S'E (Euclid VI. C.) .. D = D. 44. Let PSP' be any focal chord, and O‘its middle point. Join CO and produce it to meet the ellipse in Q. Also draw the tangent & T, and the ordinate QN. Since O is the middle point of PP, .: QT is parallel to PP','- .:: OM: MC :: QN:NC, and OM:MS:: QN:NT, .: OM:MC.MS:: QN:NC.NT, :: BCP: AC (Prop. XIII.) the locus of 0 is an ellipse, similar to the given ellipse, described upon CS as major axis. 45. Let the circle cut the minor axis in K; then since the angle BSK is a right angle, .. BK. BC= SB, A C. But if BU be the chord of the circle of. curvature at B through C, and therefore in this case the diameter, BU.BC= 2 A C (Prop. XXVI.) ... 2 BK BU, ... BK = BO, .. K coincides with the centre of the circle of curvature. 46. See fig. Prop. XI.

·PROBLEMS. Draw 80 bisecting the angle PSS', and meeting PG in 0; then O is the centre of the circle. Draw the ordinate O M at right angles to AC. Now MG: NG :: 0 G: PG, :: SG;'SG + SP, :: CS, CS + AC (Prop. XI.) Also NG:NC :: CA - CS*: AC (Prop. XII.) .. Ma: NC::(C4 – CS) CS: 40, :: CA.CS - CS: AC, and Ce: NC:: CS* : AC, .: MC: NC :: CA *CS: A C', :: OS: AC, ... MC:CS:: NO: AC, .: SM: CS:: AN: AC, and S'M: CS:: AN: AC, .. SMX SA: C8 :: ANX AN: AC, :: PN : BC (Prop. XIII.) .. SM. S'M: PN? :: CSP: CA CS. Also PN: OM :: (CA + C 8): C8, .:: SM. SM: OM :: (CA + C 8) : (CA - CS), :: AS: AS Hence the locus of O is an ellipse, whose major is SS'. Also if Cb be the semi-minor axis, Cb: CS:: AS: A'S. . . 47. Let P, P, Q, Q be the four points of intersection. Join PQ, P'Q', and let these lines or these produced intersect in 0. Draw the semi-diameters Cp,* Cp parallel respectively to PQ and P'Q. Then P0.0Q:P'0.00 :: Cp : Cpl (Prop. XXIX.)

·28 'CONIC SECTIONS, But PO.OQ='P'0. 0&from the circle, . Cp= Cp 48. Let OP, OP' be the two lines at right angles to each other, and C the centre of the ellipse in any position; then, as in Prob. 17, OC'AC" + BC", :. the locus of C is the arc of a circle whose centre is O. 49. See fig. Prop. XXIV. Draw SM at right angles to CD, and let MS and CP be produced to meet in Q. Also draw QH at right angles to the axis. Now CH: UN:: CQ : CP, :: CM: CF, CS: CG :: CN: CS:: CN: CG, :: CX: CS (Prop. XII.) :: CH= CX, .. the point Q is on the directrix. 50. For radius read diameter. If O be the centre of the circle of curvature at I, PO. PF= CD (Prop. XXVII.) .. PO: CD:: CD : PF, :: CD®: PF. CD, :: CD': AC.BC (Prop. XXII. Cor.) Hence if CD = AC.BC, .. PO = CD, ... also P.O* = AC.BC, .. PO is a mean proportional between A C and BC at the same time that CD is. Hence, if a circle be described with centre C and radius :

·PROBLEMS. 29 : a mean proportional between AC and BC, intersecting the ellipse in D, and CP be drawn parallel to the tangent at D, Pwill be the point required. 51. See Prop. XXVI. If CP= CD, then PH. CP = 2CP, .. PH = 2 CPPP, ... the circle of curvature meets the ellipse at the point P. 52. Let the linės drawn at right angles to AP, A'P meet in Q, and draw QM at right angles to AA'; then QM: AM:: AN: PN, and QM:AM :: AN: PN, .. QMʼ: AM. A'M :: AN. A'N: PN”, :: AC* : B.C? (Prop. XIII.) the locus of Q is an ellipse, similar to and concentric with the given ellipse, and having A A' for its minor axis. 53. Let C and c be the centres of the two ellipses, which intersect one another in the points P, Q, R, S. Join PQ, RS; and let these lines, or these lines produced, intersect one another in the point 0. From C and c draw the semi-diameters CU, C V parallel to PQ, and cu, cv parallel to RS; then PO.OQ:R0.08 :: CUP: CV (Prop. XXIX.) and RO. 09:RO. OS :: cu : cum (ditto.) ..: CU: OV:: cu Now if the ellipses have their major axes at right angles to one another, it is evident that since the angles UCV and uco are equal and CU, cu parallel, CU>=< CV, according as cu >< cu and .: the only case in which the proportion CV.

·80 CONIO SECTIONS. CU: CV:: cu: cv can hold godd is wlen CU and, CV'are equal. Again, if the major ages of the ellipses are parallel, the only cases in which the proportion can hold good is when the ellipses are similar, or as before when CU and CV are equal. But if the ellipses are similar, and their major axes parallel, they can only intersect in two points. Hence, whether the major axes are at right angles or parallel, we must have CU= CV. ,: PO.OQ=RO. OS. And a circle may therefore be described through P, Q, R, S. 54. Let VCV' be the fixed diameter of the circle, along Vov which CN is measured ; and draw the diameter DCU parallel to NQ; then, since NQ makes a constant angle with NP, the diameter DCD' is also fixed. Now PN = VN. V' V from the circle, .: QN = VN. V'N or QN: VN. V'N:: CD': CV. :: the locus of Q is an ellipse, of whicli CV and CD are the equal conjugate semi-diameters. The position and length of the equal conjugate diameters being now known, the ellipse may be constructed as follows, Draw ACA bisecting the acute angle formed by VCV', DCD, and BCB bisecting the supplementary angle formed by the same lines. Then A CA', BCB' :represent the directions of the major and minor axes of the ellipse respectively. Draw VM at right angles to CA, and Cv bisecting the angle ACB, and let these lines meet in v; then the point v is on the auxiliary circle.

·PROBLEMS. 81 Take CA equal to Cy, and make BO:40:: VM: M; then A C and B C are the semi-wajor and minor axes. 55. It is evident from the symmetry that the two ellipses will have a common centre, iad that that cerftre must coinoide with the point where the diameters of the parallelogram intersect. Also since the ellipses are supposed to have their major axes of the same length, their auxiliary circles will also coincide. Now since any side of the parallelogram is a common tangent to both ellipses, and by Prop. XV. the feet of the perpendiculars from the foci upon the tangent lie on the circumference of the auxiliary circle, it is evident, that the perpendiculars from the foci of one of the ellipses upon the sides of the parallelogram, will coincide with the perpendiculars from the foci of the other ellipse upon the same lines. Or, in other words, the lines joining the foci of the ellipses will be at right angles to the sides of the given parallelogram. These lines will therefore form a parallelogram equiangular to the given parallelogram. 56. Let II be the common focus of the ellipses, and S, S' their other foci. Then, since the major axes of the ellipses are equal, any point where they intersect must be equidistant from S and s". Any point, therefore, where the ellipses intersect must lie upon the line bisecting SS" at right angles, for this line contains all the points, whose distances from S and Sare equal to each other. And since this line can only intersect either ellipse in two points, it is evident that the ellipses themselves can only intersect in two points. 57. Since arc SR = arc S'R :: PR bisects the angle SPS', and coincides with the normal PQ produced.

·82 CONIO SECTIONS. Hence the triangles SPR, QP8' are similar, .: SR:PR ::88: S'P, :: CS : CA (Prop. XI.) :: C8 : CA:A0"; But PR:Q::C: NO ¢ :: AC: BC (Prop. XII.) :: SR:PQ:: CS. CA:BC". 58. Draw 80 bisecting the angle PSS', and meeting the normal PQ in 0; then Ö is the centre of the inscribed circle. Draw O M perpendicular to the axis ; then OM:PN:: 0 Q: PQ, :: SQ:SQ + SP, :: CS: CS+ CA, .. OM: CS:: PM: CS + CA, Also 2R: SP:: SP: PN (Euclid VI. B.) .: 23.1: cs. SP:: SP: CS1 CA, or 2R.: SP. SP :: CS: OS + CA. 59. Join QK, GL and let KL meet P Q in 0; then, since the angles at K and L are right angles, and the angle SPS' is bisected by PQ, :: the triangles PGL, PGK are equal in all respects. Again, since OP, PL are equal to OP, PK, and the LOPL = the LOPK, .: OL= OK, and the L POL = the L POK, .: P Q is at right angles to KL. 60. See fig. Prop. XV. Draw 8'O parallel to SP meeting YS produced in 0; then SO: SP :: SW: WI, ::'SP + SP: SP, ·PROBLEMS. 33 :: 8' 0 - SP+ SP= 240 :: the locus of O is a 'circle, whose centre is s' and radius 2 AC. w > 61. Draw the conjugate diameter CD parallel to TPt. Also draw the ordinate DR; there. TP: CD:: PN: DR, :: CR: ON (Prop. XIX. Cor.) :: CD: Pt, TP. Pt = CD. But CP. PL = TP. Pt, .. 2 CP. PL = 2 CD, = PH. CP (Prop. XXVI.) .; 2 PL PH. Again, CP. CL = CP. PL + CP, = CD? + CP = A + BC (Prop. XX. 62. Since CR bisects PQ, the tangent at R' will be parallel to P Q (Prop. XVIII.) So the tangents at P' and will be parallel respectively to QR and PR. .. the triangle rap formed by the tangents at P, Q, R' is similar to the triangle PQR. Now, since P'Q' is equal and parallel to PQ, ... the triangle r P Q the triangle PQR. So the triangle qP'R' = the triangle PQR, and the triangle p Q R' = the triangle PQR. .: the triangle par = 4. the triangle PQR. 63. See fig. Prop. XV. Draw the ordinate PN; thert PN is a common chord of both circles. D ·34 KCONIC. SECTIONS. Now L N S'J = L NY'J = L Y'NP= _ \"S' P, and LASIF LIYNOL YNP= LYSP; But LYSP= LY'S'P,. .: LNSJ = LASI, and.CS CS; :: 18 and JS' will, when produced, meet (B in the same point. 64. Draw the diameter RCR', and from any point Q on the ellipse draw Q R and QR. Also draw the diameters PCP', DCD' parallel respectively to QR and QR. Now since C is the middle point of RR', ... PCP bisects QR', .: the tangent at Pis parallel to QR', and therefore to DCD'; .:. CD is conjugate to CP. 65. See fig. Prop. XVI. If SP and SP' are in the same straight line, then the point () is on the directrix (Prop. VIII.) and the L OSP is a right angle (Prop. VI.) .. the LOMP is a right angle (Prop. XVII.) 80 also the LOMP is a right angle, ... the angles MOM' and MS'M' are equal to two right angles. But MOM' = 2 POP', .: 2T+ 0 = 2 right angles. 66. See fig. Prop. XV. A SPS = AWSS' - WSP, = SY. YY - SY. YP, = SY.PY.

·PROBLEMS 35 Now PY': S"Y: :: PY: SY, . SY. PY": SY, S'Y ::'PY : SY, or SPS : BC? :: PY: SY. So also if SY, be perpendicular to the tangent at Q, ASQS" : BC? ::-2 Y, : SY. But since the angles SPY, SQY, are complententary, .. PY: SY :: SY, :QY,; ..A'SPS : BC :: BC? : A SQS'. The angles, SPS", SQS" are evidently supplementary. "The least value of either of the angles is zero, and the greatest L SBS'. Ilence the problem is evidently impossible, unless 2 SBS' is > 2 right angles, or SBC is> a right angle, in which case CS> BC, or BC<CS. 67. Let CP be the fixed diameter; and CD the diameter which is conjugate to it in any one of the ellipses. On CP produced take a fixed point 0, and from it draw the tangent 0 Q to the ellipse to which CD belongs. Iraw QV parallel to CD, meeting CP in V, and produce PC to meet the ellipse in P'. Now (7.CO CP (Prop. XVIII.) ... CV is constant, .:. PV and P' V are also constant. Again, Q772 : P1'. PV :: CD : CP (Prop. XXI.) But CD= CP, .. Q 172 • PV. P' V, .. QV is constant. D 2 ·36 konic. SECTIONS. And since the point V is also fixed, therefore the locus of Q is a circle.whose centre is V, and. radius a mean proportional between PV and PV. 68. See fig. Prop. IV. Let A A' be one of the longer sides of the rectangle, and B' the intersection of the diagonals. Produce AB', A' B' to meet the tangents to the ellipse, at A' and A in R' and Rt Then RR' will be the side of the rectangle opposite to AA. Take any point P on the upper portion of the ellipse, and join PR, PR' meeting A A' in M and M. Also produce the ordinate NP to meet the auxiliary circle in Q. By similar triangles MM: RR :: PM : PR, :: : AN, ... NIM : MN:: AA : AN. Alsó MN: AM :: PN: AR, :: PN: BB', ... MM: AM :: PN. AA: AN.BB', :: PN. AC:AV.BC. But PN: QN :: BC: AC (Prop. XIII. ('or.) .. PN. AC= QNBC, .:MM: AM :: Qy. BC: AN. BC, :: QN: AN. So also MM' : A'M' :: QN: AN, and AN: QN:: QN: AN, ... AM: MM' :: MM : A'M'. .. 69. See fig. Prop. XXIV. Draw SH perpendicular to PG ; then PQ : PF :: SP: STI.

·PROBLEMS.
37
And if PM be the ordinate of P;*
PM : PG :: SH: SG,
:. PQ . 'PM: PF. PG :: SP: SG.
:: CA: CS. (Prop. XI.)
But PF. PG = BC (Plop. XXIII.)
.:: PQPM : BO :: CA: CS,
.. PQ varies inversely &s PM.
70. See fig. Prop. XIX.
Draw QM at right angles to A A'; then since the triangles
OMS, CNP are similar,.
.QM: SM:: CN: NP.
So also QM: SM :: CR: DR,
.. QM' : SM. SM :: GN. CR: NP. DR.
But
CN: DR :: AC: BC (Prop. XIX. Cor.)
and CR : NP :: AC: BC,
.:: ON, CR: DR. NP :: AC : BO“,
. M : SM. S'M :: AC : BC.
Hence the locus of Q is an ellipse, similar to and concentric
with the given ellipse, and which has SS' for its minor axis.

·3#
PROBLEMS ON THE HYPERBOLA.
1. LET Sand S be the centres of the given circles, and P
the centre of a circle described touching them both.
Join SP, S'P, and let these lines, or these lines produced,
meet the circles of which S and S are the centres in the
points of contact Q and Q'; then
✓ (1) If the given circles are touched both internally, or both
externally,
since PQ = PQ
:. SP w S'P=SQ
and the locus of P is one branch of an hyperbola, of which
S and s' are the foci ; but
(2) If the given circles are touched, one externally, and
the other internally,
SP + S'P = SQ + S'Q
and the locus is an ellipse.
2. See fig. Prop. VI.
Let the tangent at P meet the tangents at A and A', in
U and U'; join SU, SU'; then
since U A and UP are tangents,
.. the L USA = the LUSP. (Prop. XIV.)
So the LU'SA' = the supplement of U'SP (Prop. XIV.)
the LU'SQ,
.. the L USA is half the LASP,
and the LU' SA is half the LASQ,
.. USU' is a right angle.
.. S lies on the circle described upon U U' as diameter. So
for the point S".

·PROBLEMS. 39. 3. See fig. Prop. XVII. Join SO, S'O; then since B 02 C3-CA, .. A0*= AS. AS', .: the.. SOS' is a right angle, which proves the proposition. . 4. See fig. Prop. XVI. Let Pil' meet the conjugate axis in K; then KI": CK :: AC : AO, or KI : PN :: A C2 : BC?. But CN* – AC" : PN' :: AC : BC (Prop. X.) .:: CN - AC" = KT, or PK - KI = A C. But PK? KI” PI. PI', ... PI. PI = 40°. 5. See fig. Prop. VI. Let the tangent at A meet PT in V, and join US; then US bisects the LASP. (Prop. XIV.) and PT bisects the L SPS" (Prop, VI.) .. U is the centre of the circle inscribed in the triangle SPS', which proves the proposition. 6. PV: MN :: QN: CO :: AN. AN: A C", ... PN : AN, A'N :: MN: AC". But PN? : AN. AN :: BC : A C (Prop. X.) ... MN = BC. 7. Since CT.CN = CA' (Prop. VIII.) .::CT: CA :::04 : CN, :: CP: 0, ., AQ is parallel to PT.

·40 CONIO SECTIONS. 3 8. If P be one of the points where the ellipse and hyperbola intersect, and Sands' the common foci; then The line PT which bisects the L SPS, is a normal to the ellipse, and a tangent to the hyperbola at the point P. .. the curves intersect at right angles. 9. AR: PN :: AT : TN, and A'r: PN :: AT: TN, ... ARA'r:PN :: AT.AT: TN, :: CA-CT TS, :: CT.CN - CT: TN, :: CT. TN: TN*, :: CT : TN, ::_CT-CA: CN. TN, :: AC?: AYA'N(Prop. X.) or AR. Ar: AC? :: PN: AN.A'N, :: BCP: AC* (Prop. X.) ... AR. A'r BC AR'. A'r'. 10. See fig. Prop. XXI. Let LP, L'P't be the two tangents; then CL.Cl = CS* = CL'. Cl (Prop. XXII.) .: CL:CL' :: CL : CI, .. Ll' is parallel to L'l. 11. Šee fig. Prop. XVII. Since the L CES is a right angle, .. SE: CE:: AO: AC, :: BC: AC, But CE = AC (Prop. XVII.) .. SE BC.

·PROBLEMS. 41 12. See fig. Prop. XVII. CE:'00;: CX: CA, :: CA.: CS (Prop. II.) .. AE is parallel to so. 13. Since CP: CD' = ACH BC(Prop. XXIV., ... if AO BC2 CP CD. 14. NG: NC :: BOP: AC (Prop. IX.) if BC = AC, NG NC, ... PG CP. 15. See fig. Prop. XXV. Join QP, QP'; bisect QP, QP' in U and U', and join CU, CU'; then since QP' is parallel to CU, and QP to CU', :. CU bisects the chords parallel to CU', and CU' bisects the chords parallel to CU; .: CU and CU' are in the directions of conjugate diameters. But in a rectangular hyperbola the conjugate diameters are equal (Prob. 13), and are therefore equally inclined to either asymptote. Hence QP and QP', which are parallel to conjugate diameters, are also equally inclined to either asymptote. 16. See fig. Prop. XXIII. Since CPLD is a parallelogram, and that the diameters of a parallelogram bisect each other, .: PD is bisected by CL. 17. See fig. Prop. XVI. Join AP, and produce it both ways to meet the lines drawn through A' parallel to CR and Or in Q and Q'. Also let Q Q' meet CR and Cr in U and U'; then PU= A U', and PU' = AU (Prop. XIX.)

·12 CONIC SKCTION. But since A A' is biseeted in C, QUE AU, and Q'U' = A U', .. PQ = UN', and PQ' = UU', .. PQ = PQ'. 18. See fig. Prop. XXÝLI. SP= PI (Prop. XXXII.) = CR - AC (Prop. XVII.) So also SP = CR- BC. :. S'P' - SP=AC-BC. 19. Draw CZ parallel to SP, and D Z perpendicular to CZ; also drew SY and S'Y' perpendicular to the tangent at P; then the triangles DEC, SPY are similar. .:. DE: CD :: SY: SP, :: S'Y' : S'P(Prop. XII.) ..DE" : CD' :: SY.SY: SP. SP, :: BC? : CD? (Prop. XXXII.) ... DE= BC. 20. Let CP, CD be the given conjugate diameters. Join PD, and bisect PD in Il; and join CH. Also draw CK parallel to PD; then CH and CK are the asymptotes. Bissct the angle HCK by the line CA, and draw CB at right angles to CA; then CA and CB are in the directions of the axes. Draw the ordinate PN, and the tangent PT parallel to CD, meeting CN in T. Take CA a mean proportional between CN and CT; then CA is the transverse semi-axis.

·I PROBLEMS. 43 Also produce NP and CII to meet in R, and take BC a fourth proportional to RN, PN, and A Cpso that KN : PN :; AC: BO. Then BC is the conjugate semi-axis. P 21. Draw the ordinate QN; then, since the hyperbola is rentangular, PN* = CN°- CA' (Prop. X.), .. PN? +. AC" cr A Q = PQ". .. AQ PQ. CN, 22. See fig. Prop. XXV. Let CP and CD be a pair of conjugate diameters, which will be equal, since the hyperbola is rectangular. Let QV, drawn parallel to CD, pass through the focus S, and let it meet the hyperbola again in q. Through S draw the chord SQ'V'q' parallel to CP, meeting both branches of the hyperbola, and intersecting CD' in V'. Draw the ordmates PN, D'M at right angles to the transverse axis, and let the tangents at P and D' meet this axis in the points T and T' respectively; then CV: CP:: CS: OT. But CS. CX = CA = CN.CT.(Props. II. and VIII.) ... CS : CT :: CN: CX, Hence CV? : CP :: CN: CX? (1). Again CV': CD' :: CS: CT. But CT'. CM = CA' (Prop. XV.) = CS. CX (Prop. II.) .: CS: CT'.:: CM' : 0X, Hence CV'* : CD* :: CM'S : CX .... (2). • ·Ci : CX, :: CN 44, CONIC & criots. ... from (1) and (2), CV-C The : Cp :: CN PN : C_1? (Prop. XVI.) :: 40 : 0.1* (Prop. X.) :: CS : CA* (Prop. II.) But cs*= 2 CA*, :: CV - CI"=2 CP, .. CV - CP CP® + CV's But Qr:= CV* - CP? (Prop. XXVI.) and Q'T""? = C1?"* + CP (Prop. XXVI. Cor.) :: QV=QT, or Q9 = Q'q. 23. Since the hyperbola is rectangular, .. PN* = CN.NT (Prop. X.) .. PN: NT :: CN: PN, .. the angle PTN = the angle CPN, ... the angle TCY= the angle PCN. Hence CY: CT :: CN: CP, .:: CY. CP = CN. CT, = CA' (Prop. VIII.) .. CY: CA :: CA: CP, .. the triangles CYA, CAP are similar. 24. See fig. Prop. XVII. Let Q be the centre of the circle. Draw QF at right angles to CO, and let the latus rectum SL be produced to meet the asymptote CR in 11; then QF: QC : CA: CO, :: 'BC: CS.

·or or PROBLEMS. 45 .. QF: QF+ QC :: BC : BC + CS; QF: BC :: AC: B'S, (1) where, is the focus of the conjugate hyperbola(Art. 56). Again, LS? : C 82 CA? :: BC" : AC", (Prop. X.) LS: BC :: BC' : 4C", .. LS: BC :: BC AC, and HS : BC :: CS : AC, .:. HL. BC :: CS-BC: :: BS, : AC.. (2). Now B-S,. B'S, CS = AC, .:: AC: B'S, :: BS, : AC. Hence QF: BC :: HL : BC, .: QF=HL. AC, . CB, 25. See fig. Prop. XIX. Let the chords be drawn parallel to CR; then CH.IIR:RQ.QR' :: CR: RR', or CH.CH : PL :: CR’: 4 RV" (Prop. XX.) :: CL: 4 PL, 4CH.CH = CL = 4CK", ... CH.CH' = CK'. 26. RK: RP :: P'K':P'R', but PR = P'R' (Prop. XIX.) :. RK = P'K', so also R'K' = PK. 27. Draw RM perpendicular to A A' produced; then RM: PN :: AM: AN, and RM:QN:: A'M: A'N, ..R12 : PN.QN:: AA, AJI: ANAN, ·46 ECTIONS. TONS. but PNON = AN. A'N, .. RM = AM. A'M. the locus of R is a rectangular byperbola, of which A A' is the transverse axis. 28. See fig. Prop. XXyI. Let P be one of the points in which the hyperbolas in the figure intersects the hyperbola whose axes are in the directions of CL and Cl'; then, since the hyperbolage are rectangular and equal, it is evident that CP bisects the angle RCN. Now the angle CPT = the angle PCD, = 2. the angle PCI, = the angle RCN, = half a right angle, the angle between the tangents to the hyperbolas at P is a right angle. : 29. Join A'P, and draw PK at right angles to A'P, meeting the transverse axis in K'; then since PN : AN. A'N :: BC' : AC (Prop. X.) A'N.NK: AN. AN :: BC: AG?, or NK: AN:: BC : AC, :: QP: AQ, :: NQ is parallel to PK. ... also QI drawn at right angles to QN is parallel to A'P. Hence AH: HA' :: AQ:QP. 30. Let SH be the given base; and P, Q the points of trisection of a segment described upon it. It is evident that P Q is parallel to 8H. · Bisect SH in X, and draw XM perpendicular to PQ, bisecting it in M; then SP= PQ = 2 PM ·PROBLEMA. 47 .:: the locus of P is a hyperbulą whose focus is 8, and directrix XM. 31. Since the triangles 8Cs, TCt*are equal (Prop. XXII.) .: also the triangles TVS, V8 are equal, and the 4 S VT is equaloto the Z 8Vt, .:18: Vå :: Vt : VT (Euclid, VI, 14. 32. See fig. Prop. XII. Let S'Y' meet the auxiliary circle in 2'; and draw Z'M' perpendicular to SP, then SM': S'Z' :: SY': SP, .: S'M'. $P = SZ'. SY', = BC (Prop. X11.) .:: ZM is the chord of contact of the pair of tangents drawn from P to the circle whose centre is S, and radius BC. Again, since CY produced passes through Z, and CY is parallel to S'P, ... the L CZ'M' is a right angle, .:: ZM' touches the auxiliary circle. 33. (1) Let ( be between A and B; and let the tangents AQ, BR meet in P; then AP n PB AQ BR, AC BC. Therefore the locus of P is a hyperbola, of which A and B are the foci; unless AC is equal to BC, when the locus is a straight line bisecting A B at right angles. (2) Let C be on AB produced, and let the tangents AQ, BR as before meet in P; then, AP + BP = AQ + BR, > AC + BC, ... the locus of P is an ellipse.

·48 •CUNIG SĘCrios. 34. Draw* QM perpendicular to A A'; then M: AM :: PN: AN, and QM: AM :: P'N: A'N, .: QM:AM.AM ::'PN : AN. A'N, :: BC*: AC?. .. the locus of Q is a hyperbola, of which AC and BC are the semi-axes. 35. Since PF. CD AC. (Prop. XXV. Cor.) and PF. PG BC". .. CD: PG :: AC: BC;. hence, when the hyperbola is rectangular, PG = CD = PL (Prop. XXIII.) and PL = Pl (Prop. XIX. Cor.) .. the angle LGl is a right angle. 36. Let A be the common vertex, and NPQ a common ordinate to the ellipse and parabola ; then PN: AN. AN:: BC: AC, or denoting the diameter of the circle of curvature at A .. by 2 R. PN : AN. A'N:: R. AC: A C (Prop. XXVI. Ellipse) :: R: AC, :: 2h . AN : 2 AC. AN, .:. PN : 2R. AN :: AN. AN: 2 AC. AN: :: AN: 2 AC! But QN = 4 AS. AN, 2R :: AN (Prop. XIX. Parabola) .: PN : QN :: AN: 2 AC, Now in the ellipse A' N is less than 2 A C. .. PN is < QN.

·ÞROBLEMS.
Again, if NP is an ordinate of the hyperbola, the søme
demonstration applies, except that
•A'N is > 2 AC,
and .: PN> QN.
37. Produce RQ to meet the asymptote in and uraw
PL parallel to CR', meeting CR in L; then
since Rr is bisected in (Prop. XIX. Cor. 1)
:: 2QK = CR
= CL + RL
But RL = P'H',
since RP = R' P' (Prop. XIX.)
.. 2QK CL + P'H',
PH + P H.
38. Draw PH, PH', parallel to one asymptote, and PK,
PK' parallel to the other.
Let PH, P'K' meet in Q', and P'H', PK' in Q; then
PH.PK Ρ Η' P'K' (Prop. XXI.)
.. PH : P'K' :: PH' : PK,
or CK: KQ :: CK : K'Q',
.Q Q passes through C.
39. Draw PH, PK parallel to the sides CF, CE of the
rectangle; then
4PH.PK = CE. CF,
Ph.
and is therefore constant.
Hence the locus of P is a rectangular hyperbola, of which
the sides of the rectangle are asymptotes.
40. Let the tangent at P meet the asymptotes CM, CN in
L and I.

·50 'CONIC SECTIONS. Join MIŃ; then PMand PN are tangents' to the ellipse (Art. 36, Def.) .: the tangent at Q is parallel to MN (Prop. XVIII. Ellipse.). Also, since Ll is bisected in P (Prop. XIX. Cor. 1) .. CL and Clere bisected in M and N, .. Lois parallel to MN, the tangentsat P and Q are parallel. 41. NP. NQ = AN. AN, .- PA : NP. NO :: BC" : AC?, or PN: ON:: BC? : AC?, :: BC:B'C', if 'B' C bc a third proportional to BC and AC, .: ON : PN? :: B'C' : BC", and PN : AN'AN:: BC: AC, .. QN; AN. AN:: B'C' : AC", ... the locus of Q is a hyperbola, of which AC and B'C are the transverse and conjugate semi-axes. 42. See fig. Prop. XXV. Let PP' be the diameter, and any point on the curve. Join PQ, P'Q, and draw CU parallel to P' Q, bisecting PQ in the point U. Then if CU' be drawn parallel to PQ, CU and Cu will be conjugate, diameters, and therefore (see Prob. 15) will make equal angles with CR, as also with CP and CD. Hence LQPP - LQP'P= LUCP - LUCP, = LUCP - LOCD, = L PCD < 43. Let TPt be a common tangent to the hyperbola and any one of the elliptic quadrants APB, and let it meet the asymptotes CA, CB in Tana t.

·PROBLEMS. 51 Draw the ordinates PN, Pn parallel respectively to Ct and CT; then since TPt is bisected in P (Prop. XIX. Cor. 1) i. CT is bisectpal in N. But from the ellipse A C? ON. CT, .. AC = 2 ON. So also B02 2Cn?, .. AC.: BC? :: CN : Cn", AC: BC :: ON: Cn, .:. AC.BC: BC :: CN. Cn : Cn. BC 2 CN. Cn, = 208. (e rop. XXI.) .. AC . 44. Produce OP to meet CD, the diameter conjugate to CP, in F'; then OP:P :: CP: PF, ... OP. P1 PC.CP, CP, .. O is the centre of the circle of curvature at : (Prop. XXIX.) 45. See fig. Prop. XXV. Let Cl and CL be conjugate diameters of an ellipse, with C as centre, described touching the hyperbola, of which Cl and CL are asymptotes, in P. Draw the common tangent IPI, and from the point L draw LDI', touching the conjugate hyperbola in D;. then CP and CD are conjugate diameters of the hyperbola, and PD is parallel to Il'. (Prop. XXIII.) E 2 ·52 CONIC SECTIONS. Join PD, meeting, CL in M; then sinee PD is a parallelogram, ...MD PM. Hence, since CL and Cl' are conjugate diameters of the ellipse, .. D is a point on the ellipse; and, since the tangent to the ellipse at P is parallel to CD, in the tangent to the ellipte at D is parallel to CP, .: LD touches the ellipse at D, or the ellipse' and hyperbola have a common tangent at D, and therefore touch one another. Also CP and CD are conjugate diameters of the ellipse, as well as of the hyperbola. 46. Let Q, Q' be the points where the common tangents QP, Q'P' touch the hyperbola, and P, P' the points where they touch the ellipse. Through 0, the point of intersection of QP and Q'P', draw ('O, meeting PP' and QQ' in U and V respectively. Join CQ, CQ' meeting the ellipse in q, q'; then since QQ', PP' are bisected in V and U,. they are evidently parallel. Now, since a line drawn through q, parallel to PP and QQ', will be bisected by CUV, whether it be terminated by the ellipse or C'Q', it is evident that qa' is parallel to QQ. Hence CQ: CQ' :: Cq:Cq'. an ellipse similar to the given ellipse, and having also C for its centre, can be described through the points Q and Q'. 47. Let PO, PO' be the radii of curvature at P and P' respectively; and let PF, P'F' be the perpendiculars from P Andur, upon the diameters CD and CD', conjugate respectively and CP'; then ·PROBLEMS. 53 PQ. PF = CD (Prop. XXIX.) .. PO.PF.CD = ÇOS, or PO.AC. BC = CD: (Prop. XXV: Cor:) So also P'O'.AC.BC CD'S .:: PO: P'O' :: CD CD's, :CP. CP's (Prob. 18), since the hyperbola is rectangular. 48. Let R and S be two of the points of intersection on the same side of PQ;'and let PQ intersect the hyperbola in p and q. Draw O V bisecting PQ in V; then since PP Qq (Prop. XIX.) ... l' is also the middle point of PI. Now, since the chords which are drawn parallel to P Q are bisected both in the ellipse and hyperbola by OV, it is evident that the chord drawn from R, parallel to PQ, must meet the ellipse and hyperbola in the same point. Hence RS is parallel to P Q. So also, if R', S' be the points of intersection of the hyperbola and ellipse on the other side of PQ, R'S' is parallel to PQ. l'roduce PR to meet O V in T, and produce TS to meet PQ in Q'; then if RS meet OV in M, Q'V: TV :: SM : SM, :: RM: TM, :: PV: TV, :: QV:TV, :: Q and coincide, and TS passes through the point Q. Hence, since the tangents to the hyperbola at R and s meet OV in the same point (Prop. XXVII. Cor.), and PR and ·54 CONIC SECTIO S. es have also been proved to intersect op OV, it is evident that, if PR is the tangent at R, QS must also be the tangent at S. Lastly, whether U be the point where RQ or PS intersects 01", MU: UV :: RJ: PV, .: RQ and PS meet on the line OV, which bisects P'Q. 49. Let the points A and B be on the same branch of the hyperbola. Let A Q and BP intersect in R, join CR, and produce CR to meet dB in 0; then, since the angles at Pand Q are right angles, and the three perpendiculars from the angles of a triangle on the opposite sides meet in a point, .:: CR is at right angles to A B. Now by similar triangles CO:0A :: BO:OR, ... CO.OR = OA.BO. Hence, by the converse of the Cor. of Prop. XXVIII., since the hyperbola is rectangular, C is also a point on the hyperbola, but situated on the opposite branch to that on which AB is drawn, and on which C moves.

·. PROBLEMS ON THE SECTIONS OF THE CONE. 1. See fig. Prop.'I. Let C be the centre of the sphere instribed in the cone, and touching the plane of the parabolic section in the focus S. Join CA, CE; then since AS is parallel to Oe, A is evident that the LACO is a right angle Now SA:AO :: AE: AO. But since AE: AC :: AC :10, .. AE:A0:: AC : AO. Hence SA:AO :: AC? : A0%, .:: the ratio of SA to A O is independent of the position of the point A, and depends only upon the vertical angle of the cone. The foci therefore of all parabolic sections made by planes perpendicular to the plane of the paper, will be upon a straight fine drawn through 0; and the foci of all the parabolic sections that can be made by any planes, upon the surface of the cone formed by the revolution of this fine round the axis of the given cone. 2. See fig. 2, Prop. I. CS:CA :: SA: AX, :: AE: AX Now if the ratio of A E to 44 X is fixed, since A EX is a fixed angle, and AE is dess than AX, and A YE is therefore less than a right angle;

·56 CONIO SBCTIONS. the LAXĚ is fixed, and can have only one value (Euclid, VI. 7.), .. also the L SA is fixed. Draw AC to the centre of the inscribed sphere bisecting the angle 8A0; "then the Jungle CAE is a fixed angle. Now SA: AQ :: AE: AO, and the ratio of AB to AO is compounded of the ratios of AE: AC and AC: 40; but since the 4 CAE is fixed, the ratios of AE: AC, and AC:40 are both fixed ratios. Hence SA : A0 is a fixed ratio. Therefore, as in the preceding problem, the locus of S is ca cone, havirg ( for its vertex. In the sathe manner it may be shown that the other focus • 8' will always lie on another cone which has also O for its vertex. . 3. See fig. 2, Prop. I. Let A A' be the major axis of one of the ellipses. Since the cutting planes are parallel, the major axes of the elliptic sections will be parallel also. Hence, if O be joined with the middle point of A A', the line so drawn will also pass through the centres of all the other elliptic sections. If through this line a plane be drawn perpendicular to the plane of the paper, the extremities of the minor axes will all lie on the two lines in which this plane intersects the surface of the cone. 4. See fig. 1, Prop. I. The latus reolum = 4AS, and in Prob. 1 it has been shown that AS bears a ratio to A 0, depending only upon tko vertical angle of the cone. Hence the latus rectum varies as AO ·PROBLEMS. 5. See fig. 3, Prop. I. The angle KOL represents half the angle Wetween the asymptotes of the section made by a plane parallel to KOL. Now, in the rectangular hyperbola, the angle between the asymptotes is a right angle, and the greatest value which KOL admits of is half the vertical angle of the cone Hence, if the vertical angle of the cone is less than a right angle, it will be impossible to cut it in sach a manner that the section may be an equilateral hyperbola. When the vertical angle is a right angle any section made by a plane parallel to a plane through the axis will intersect thre curve in a rectangular hyperbola. When the angle R Or is greater than a right aigle, bisect OR in V, and draw VZ at right angles to SR, making VZ equal to 0 V, and join OZ; then OR? = 2 IZ? With centre 0, and radius o Z, describe a circle, cutting Rr in L; then KL = OK - OL”, = OR - OL, = 2 OL- OL', = OL. Hence the angle KOL is half a right angle, and therefore any plane parallel to KOL will intersect the cone in an equilateral hyperbola. 6. When the cutting plane is parallel to 01 and Oi, the perpendicular from 0 on the cutting plane is the conjugate semi-axis of both hyperbolas. The latera recta are therefore inversely proportional to the transverse axes. (Art. 48.) Also O1, Od, and OD, Och are parallel to the asymptotes of the hyperbolas in the cones whose axes are respectively. 0 1 and Oi. Let Oa, ab, measured alohg and perpendicular to 01, represent the semi-axes of the one hyperbola, and Oa', a'b',

·..
58
CONIC SECTIONS.
measured along and perpendicular to Oi, the semi-axes of
the other hyperbola,
it
Then
Oa : ab :: '01 : li,
and
a'l' : Oa ::.()1 : Oi;
since
abw= a'b',
oa da :: OI : 01?
:: the latera recta are as Oi* : 0r.
Next, when the cutting planc is perpendicular to 01 and
Oi, the transverse axis of the hyperbola is equal to the major
axis of the ellipse.
Let 'AA' be the common axis (sce fig. 3, Prop. L.), Ctke
common centre; then, if Cuu' be drawn parallel to (7,
meeting on, Od' in u and u'; then
the latera recta are as CU. CUM : Cu. Cu' (Prop. I.)
as (2 : 01

·59
APPENDIX.
1. The following is an independent solution of Prob. 26 of
the Parabola.
'Join AP, 4%, and let them meet' in p and q, the line
drawn through at right angles to the axis; then since
Dp : PM :: A0 : AM,
and PM : 44S :: AM : PM,
... Op : 4 AS :: AO: PM.
So also 09 : 4 AS :: AO: QN,
:: Op : 09 :: QN: PM,
:: 20: PO,
.: P Q is parallel to Pq,
.: Ap : AP:: AQ : A4,
.:: A(): AM :: AM: AO,
... AO = AM.AN.
2. The following proof of the more general property, of
which Prob. 44 of the ellipse is a particular case, is deserving
of notice.
Let O be any point. It is required to find the locus of the
iniddle points of chords drawn through 0
Join OC, and produce it to meet the ellipse in K and K'.
Draw any chord Pop, and let Q be its middle point, and
through K draw the chord KR parallel to P Op. Produce
CQ to meet KR in r; then KR is bisected in r.
Hence
OQ: Kr :: OC : C-K,
.:: 0 Q : KR':: 00 : KK;',
i.e. O Q.bears a constant ratio to KR,
• the locus of Q is an ellipse similar to the given ellipse.

·OVDON
CLAY, AON, AND TAYLOR, PRIVTERS,
HREAD NTRFET HIIL.

·Cambridge,
And 28, HENRETTA STREET, COVENT GARDEN,
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·This Series of CAMBRIDGE CLASS-Books FOR THE USE OF SC110013 AND COLLEGES, which have been issued at interrals during the last thr years, is intended to embrace all branches of Education, from the most clementary to the most adranced, and to keep pace with the latest discoveries in Science. Of those hitherto published the sale of many thousands is a sufficient 'indication of the manner in which they have been appreciated by the public. 4 SERIES of a more Elementary character is in preparation, a list of which will be found at the back of this Catalogue.

·CAMBRIDGE CLASS BOOKS FOR SCHOOLS AND COLLEGES. WORKS by the Rev. BARNARD SMITH, M.A. Fellow of St. Peter's College Cambridge. 392 pp. Arithmetic and Algebra 2. Arithmetic In their Principles and Applications. For the Use of Schools. With nuncrous Eramples, systemat New Edition (1862) 348 pp. ically arranged. Crown 8vo. strongly bound in cloth, 48. 6d. Answers to all the Questions. Eighth Edition. 696 pp. (1861). Crown 8vo. strongly bound in cloth. 10s. 6d. 3. Key to the above, containing Solutions to all the Questions in the The first edition of this work was published latest Edition. Crown 8vo. cloth. in 1874. It was primarily intended for the use of students at the Unicrsities, Second Edition. 88. 6d. and tor Schools which prepare for the To meet a widely expressed wish, the L'niversities. It has however been found ARITIIMETIC was published separately to meet the requirementsofa inuch larger from the larger work in 1854, with so much Cluss, und is now catensively used in alteration as was necessary to make it quite Schools and Collegex both at home and independent of the ALGEBRA. It has now in the colonies. It has also been found a large and increasing sule in all classes of of gweat service for students preparing Schools at home and in the Colonies. A very for the Miupla-Class and Civil AND copious collection of Examples, under each MILITARY SERVICE EXAMINATIONS, froin rule, has been enubodied in the work in a the care that has been taken to elucidate systematic order, and a Collection of Miscelthe principles of all the Rules. Testimony laneous Papers in all branches of Arithmetic of its efecllence has been borne by some have been appended to the book. of the highest practiqul and theoretical authoritier; of which the following from 4. Exercises in Artththe late DEAX PEACOCK may be taken do a specimch. metic. 104 pp. Crown 8vo. (1860) "Mr. Smith's Work is a most useful pub 2s. Or with ANSWERS, 2s. 6d. Also lication. The Rules are stated with great sold separately in 2 Parts 18. each. clearness. The Examples are well selected . Answers, 6d. and worked out with just sufficient detail without being encumbered by too minute The EXERCISES have been published in explanations; and there prevails through order to give the pupil examples of every out it that just proportion of theory, and rule in Arithmetic, and a great number have practice, which is the crowning excellence been carefully compiled from the latest Unie of an elementary work." versity and School Examination Papers, ·CAMBRIDGE CLASS BOOKS WORKS by ISAAC TODHUNTER, M.A. F.R.S. Fellow and Principal Mathematical Lecturer of St. Jok.n's College, Cambridge. 1. Algebra. Spherical Trigonometry For the use of Colleges and Schools. For the Use of Colleges and Schools.. 112 pp. Crown 8vo. (1869). Strongly bound in cloth. 48. 6d. Third Edition. 542 pp. (1862). This work is constructed on the same plan as the Treatise on Plane Trigonometry, to Strongly bound in cloth. 78. hd. which it is 'ntended as a sequel. Considerable labour hus been expended on the This work contains all the propositions tecurte, and the Examples, which have been which are usually included in elementary chiefly selected from l'niversity and College treatises on Algebri, and a large number l'apers, have all been carefully verified. of Examples for Erercise. The author has sought to render the work easily intelligible 4. The Integral Calculus to students without impairing the accuracy of the demonstrations, or contracting the And its Applications. limits of the subject. The Examples have been selected with a view to illustrate every With numerous Examples. part of the subject, and as the number of Second Edition. 312 pp. (1862). them is about swteen hundred and fifty, it is hoped this will supply ample exercise Crown 8vo, cloth. 108. 61. for the student! Each set of Examples has In writing the present Treatise on the veen carefully arrangedl, commencing with Integral Calculus, the object has been to very simple exercises, and proceeding grudu- produce a work at once elementary and comally to those which are less obvious. plete-adapted for the use of beginners, and Authcient for the wants of advanced students. In the selection of the proportionn, und in the mode of cutablishing them, the author has endeavoured to exhibit fully and clearly 2. Plane Trigonometry. trate all their most impurtant results. In order that the ntudent may find in the For Schools and Colleges. volume all thut he requires, a large collection of Examples for exercise has been appended to the different chapters. Second Edit. 279 pp. (1860). Crn. 8vo. 6. The Strongly bound in cloth. Differential Calculus. The design of this work has been to render With numerous Examples. the subject intelligible to beginners, and at Third Edition, 398 pp. (1860). the same time to afford the student the Crown 8vo. cloth, 108, 6d. opportunity of obtaining all the information which he will require on this branch of This work is intended to exhibit a com- Mathematics. Each chapter is followed by prehensive view of the Differential ('alculus a set of Examples ;, those which are entitled on the method of Limits. In the more ele- Miscellaneous Examples, together with a mentary portions, explanations bave been few in some of the other sets, may be ad- Riven in consideravle detail, with the hope vantageously reserved by the student for that a reader who is without the assistance exercise after he has made some progress of a tutor may be enabled to acquire a comin the subject. As the Text and Examples petent acquaintance with the subject. More of the present work bave been tested by con than one investigation of a theorem has been siderable experience in teaching, the hope is frequently given, because it is believed that entertained that they will be suitable for im. the student derives advantage from viewing parting a sound and comprehensive know the same proposition under different aspects, ledge of Plane Trigonometry, together with and that in order to succecd in the examinareadiness in the application of this know tions which he may have to undergo, he ledge to the solution of problems. In the should be prepared for a considerable variety Second Edition the hints for the solution of in the order of arranging the several branches the Examples have been considerably in of the subject, und for a corresponding creased. tariety in the mode of demonstration. 58.

·FOR SCHOOLS AND COLLEGES. WORKS by ISAAC TOPHUNTER, M.A. F.R.S.--continued. 6. Analytical Statics. 9. Plane Co-Ordinate Geometry With numerous Escamples. AS APPLIED TO THE STRAIGHT LINE Socond Edition. 330 pp. (1858). AND THE CONIC SECTIONS ; Crown 8vo. cloth. 198. 6d. With numerous Examples. In this work will be found all the pro Third and Cheaper Edition. positions which usually appear in treatises on Theoretical $tutics. To the different Crown 8vo. cl. 326 pp. (1862). 78. 6d. chapters Examples are appended, which have been selected principally from the This Treatise exhibits the subject in a University and College Examinatian Papers; simple manner for the benefit of beginners, these will furnish ample exercise in the ap- and at the same time includes in one volume plication of the principles of the subject. all that students usually require. The Ex. amples at the end of each chapter will, it is hoped, furnish sufficient exercise, as they have been carefully selected with the view 7. AN ELEMENTARY TREATISE ON THE of illustrating the most important points, and have been tested by repeated experience with pupils. In consequence of the demand been originally anticipated, a large number With a Collection of Examples. of copies of the Third Edition has been printea, and a considerable reduction effected Crown 8vo. cloth. 73. 6d. in the price. OF THE This treatise contains all the propositions which are usually included in elementary treatines on the Theory of Equations, to- 10. History of the Progress gether with a collection of Examples for exercise. It may be read by those who are familiar with Algebra, since no luigher knows Calculus of Variations ledge is assumed, except in Arts, 175, 267, 308-314, which may be postponed by those who are not acquainted with De Moivre's During the Nineteenth Century. Theorem in Trigonometry. This work may 8vo. cloth. 632 pp. (1861). 128. M fact be regarded as a sequel to that on Algebra by the same writer, and accordingly It is of importance that those who wish to the student has occasionally been referred cultivate any subject may be able to ascerto the creatise on Algebra for preliminary tain what results have already been obtained, information on some topics bere discussed. and thus reserve their strength for difficulties which have not yet been conquered. And those who merely desire to ascertain the present state of a subject without any 8. EXAMPLES OF purpose of original investigation will often find that the study of the past history of that subject assists them materially in Outaining Analytical Geometry of a sound and extensive knowledge of the condition which it has attained. The Author Three Dimensions. has endeavoured in this work to ascertain distinctly what has been effected in the 76 pp. (1858). Crown 8vo. cloth. 48. estimate of the manner in which it has been Progress of the Calculus, and to form some effected : accordingly, upless the contrary A collection of examples in illustration of is distinctly stated, it may be assumed that Analytical Geometry of Three Dimensions any treatise or memoir relating to the Calhas long been required both by students and cylus of Variations which is described in this teachers, and the present work is published work has undergone thorough examination with the view of supplying the want. and study.

·6 CAMBRIDGE CLASS BOOKS. By J. H. PRATT, M.À. By Reo. S. PARKINSON, B.D. Archdeacun de Calcutt., late Fellow of Fellow and Prælector of St. John's Coll. Gonville and Caius College, Cambridge. • Cambridge. A Treatise ota 1. Elementary Treatise Attractions, on Mechanics. Ta Place's Functions, and the Figure With a Collection of Examples. of the Earth. Second Edition. Crown 8vo. 126 pp. Second Flition. 345 pp. (1861). Crown 8vo. cloth. 98. 6d. (1861). cloth. 58. 6d. The Author has endeavoured to iender In the present Treatise the author has the present volume suitable in a Munual for endeavoured to supply the want of a work the junior classes in 'niversities and the on a subject of great importance and high higher classes in Schools. With this object tions and the calculation of the figure of there have been included in it those portions the Earth by meals of his remarkable analy; veniently investigated without the Differen: of theoretical Mechanics which can be consis. Yo student of the higher branches of tial Calculus, and with one or two sport Physical Astronomy should be ignorant of La Place's anal-sis and its resulta calculus," exceptions the student is not presymer to says Airy, the most singular in its nature Matncmatics beyond the elements of Algebra. and the most powerful in its application that has ever appeteed." A collection Grometry, and Trigonometry. of l'roblems and Examples has been added. cbiefly taken from the Senate-House and By P. G. TAIT, Y.A., and College Examinatiga Paperwhich will, it is trusted, be found useful as an excrcinc 1. J. STEELE, B.A. for the student In the Second Edition Late Fellows of St. Peter's Coll. Cambridge. wererul additional propositions have been in- Dynamics of a Particle. corporated in the work for the purpose of rndering it more complete, and the Collec- With numerous Examples. tion of Examples and Problems has been largely increased. 304 pp. (1856). Cr. 8vo. cl. 108. 6d. In this Treatise will be found all the ordi- 2. A Treatise on Optics. nary, propositions connected with the Dynamics of Particles which can be convenienti, 304 pp. (1859). Crown 8vo. 10s. 6d. deduced without the use of D'Alembert's Principles. Throughout the book will be The present work may be regarded as a found a number of illustrat ve Examples new edition of the Treatise on Optics, by introduced in the text, and for the most part the Rer. W. X. (iriffin, which being some completely worked ont; others, with occa- time ago out of print, was very kindly and sional solutions or hints to assist the student liberally placed at the dixporul of the author. are appended to each Chapter. The author has freely used the liberty accorded to him, and has rearranged the matter with considerable alterations and additions By the Reo. G. F. CHILDE, M.A. --especially in those parts which required Mathematical Professor in the South African more copious explanation and illustration to College. render the work suitable for the present А Singular Properties of collection of Examples and Problems has been appended, which are sufficiently nuthe Ellipsoid merous and varied in character to afford an Useful exercise for the student: for the And Associated Surfaces of the nth greater part of them recourse has been had Degree. to the Examination Papers set in the Uni. 152 pp: (1861). 8vo. boards. 108. ou versity and the several Colleges during the last twonty years. As the title of this volume indicates, its Subjoined to the copious Table of Contents object is to develope peculiarities in the the author has venturned to indicate an ele- Ellipsoid ; and further, to establish analo- mentary course of reading not unsuitable gous properties in unlimited congenetic for the requirements of the First Three series of which this remarkable surface is Days in the Caribridge Senate House Ex- A constituent. aminations.

·1 FOR SCHOOLS AND COLLEGES. By J. B. PLEARP M.A. BY GEORGE BOQLE, D.C.L. Fellow and late Mathematical Lecturer of Professor of Mathematics in the Queen's Clare College. University, Ireland. 1. Differential Equations Elementary Hydrostatios 468 pp. (1859). Crown 8vo. cloth. 148. With numerous Examples and Solutions. The Author has endeavoured in this trea- Second Edition. 156 pp. (1857). tise to convey as complete an account of the present state of knowledge on the subject of Crown 8vo. cloth. 68. 6d. Differential Equations as was consistent with the idea of a work intended, primarily, " An excellent Introductory Book. The for elementary instruction. The object has definitions are very clear; the descriptions been first of all to meet the wants of those und explanations are sufficiently full and who had no previous acquaintance with the intelligible; the investigutions are simple subject, and also not quite to disappoint and scientific. The examples greatly en-others who might seek for more advanced hance its value."--ENGLINN JOURNAL OF information. he earlier sections of each EDTCATION. chapter contain that kind of matter which This Edition contains 147 Examples, and has usually been thought suitable for the solutions to all these examples are given at either to an account of recent discovery, or beginner, while the latter ones are devoted the end of the book. to the discussion of such deeper questions of principle as are likely to present themselves to the reflective studeng in connection with the inethods and processes of his previous course. By G. B. AJRY, M.A. Astronomer Royal. 2. The Calculus of 1. Mathematical Tracts. Finite Differences. On the Lunar and Planetary Theories, Crown 8vo. cloth. 248 pp. (1860). 108. 6d. Figure of the Earth, the Undulatory Theory of Optics, gc. In this work particular attention has been paid to the connexion of the methods with those of the Differential Calculus & con- Fourth Edition. 400 pp. (1858). nexion whica in some instances involves 8vo. 158. far more than a merely formal analogy. The work is in some measure designed as a sequel to the Author's Treatise on Diffe- 2. Theory of Errors of rential Equations, and it has been composed on the same plan. Observations .By J. C. SNOWBALL, M.A. And unte Combination of Observations. Late Fellow of St. John's Coll. Cambridge. 103 pp. (1861). Crown 8vo. 68. 6d. Plane and Spherical Trigonometry. • In order to spare astronomers and observers iu natural philosophy the confusion With the Construction and Use of Tables ferring to the ordinary treatises embracing of Logarithms. both branches of Probabilities, the author Ninth Edition. 240 pp. (1857). Crown has thought it desirable to draw up this 8vo. 78. 60. work, relating only to Errors of Observation, and to the rules derivable from the In preparing a new edition, the proofs of consideration of these Errors, for the Com- some of the more important propositions have bination of the Results of Observations. been rendered more strict and general, and a The Author has thus also the advantage considerable addition, of more than two hunof entering somewhat more fully into several | dred Examples, taken Principally from the points of interest to the obseryer, than can questions in the Examinations of Colleges and possibly be done in a General Theory of the University, has been made to the colleo- Probabilities, tion of Examples and Problems for practice.

·8 CAMBRIDGE CLASS BOOKS. By W. H. DRËT, M.A. By EDWARD JOHN ROUTH, M.A Second Master of Blackheath School. Fellow and Assistant Tutor of St. Peter's Geometrical Treatise on College, Cambridge. Conic Sections. Dynamics of a System With a copious Collection Examples. of Rigid Bodies. Second Edition. Crown 8v8. cloth. 48. 6d. With grumerou. Examples. In this work the subject of Conic Sections 336 pp. (1860). Crown 8vo. Cioth. has been placed before the student in such 108, 6d. a form that, it is hoped, after masteringa the elements of Euclid, he may ind it an easy and interesting continuation of his geometrical studies. With a view also of CONTENTO: Chap. I. Of Moments of Inrendering the work a complete Manual ofertia. - II. D'Alembert's Principle. - III. what it required at the Universities, there Motion about a Fixed Axis.-IV. Motion in have been either embodied into the text, or Two Dimensions.-V. Motion of a Kikid inserted among the examples, every book- Body in Threr Dimensions.-VI. tion of work question, problem, and rider, which a Flexible String.-VII. Motion of a system has been proposed in the Cambridge exami- of Rigid Bodies.-VIII. Of Impulsive Forces. nations up to the present time. -IX. Miscellaneous Examples. The numerous Examples which will delen the Pro- chiefly selected froni thc Examination Papers found at the end of sich chapter have been blems in Drew's Conic bridge during the last few yeurs. set in the University and Colleges of Cam- Sections. Crown 8vo. cloth. 48. 6d. Solutions to By G. H. PUCKLE, M.A. By R. D. BEASLEI, M.A. Principal of Windermere College. Ilead Master of Grantham School. Conic Sections and Plane Trigonometry. Algebraic Geometry. AN ELEMENTARY TREATISE. With numerous Easy Examples Progressively arranged. With a numerous Collectum of Second Edition. 264 pp. (1866). Examplcs. Crown 8vo. 78. 6d. 106 pp. (1858), strongly bound in This book has been written with special cloth. 38. 6d. reference to those difficulties and misapprehensions which commonly beset the student This Treatise is specially intended for use when he commences. With this object in in Schools. The choice of matter has been view, the earlier part of the subject has chietty guided by the requirements of the been dwelt on at length, and geometrical three days' Examination at Cambridge, with and numerical illustrations of the analysis the exoeption of proportional parts in logahave been introduced. The Examples ap-. rithms, which have been omitted. About pended to each section are mostly of an four hundred Examplos have been added, elementary description. The work will, it painly collected from the Examination is hoped, be found to contain all that is re- Papers of the last ten years, and great pains quired by the upper classes of schools and have been taken to exclude from the body by the generality of students at the Uni of tbe work' any which might dishearten a versities beginner by their difficulty.

·FOR SCHOOLS AND COLLEGES. The J. BROOK SMITA, M.A. Cambridge Year Book Arithmetic in Theory St. Johns College, Cambridge. AND UNIVERSITY ALMANACK and Practice. For 1862. For Advanced Pupils. Crown 8vo. 228 pp. price 28. 6d. Part. Crown 8vo. cloth. 38. 61. The specific features of this annual pub on Arithmetic, in which the Author has This work forms the first part of a Treatise lication will be obvious at glance, and its value to toachery engaged in preparing explain in a full and satisfactory manner all endavoured, from very simple principles, to students for, and to parents who are sending the important processes in that subject. their sonk to, the University, and to the The proofs have in all cases been given in public generally, will be clear 1. The whole mode of proceeding in enter does not think that recourse ought to be a form entirely arithmetical: for the author ing a student at the University and at any had to Algebramuntil the arithmetical proof particular l'ollege is stated. has become hopelessly long and perplexing. 2. The course of the studies as regulated At the end of every chapter several exby the University examinations, the manner amples have been worked out at length, in of these examinations, and the specific sub- which the best practical methods of operajocarun times for the year 1862, are given. tion have been carefully pointed out. 3 .complete account of all Scholarships and Exhibxions at the several Colleges, their value, and the means by which they are By N. M.'FERRER, M.A. fained. Fellow and Mathematical Lecturer of Gon 4 A brief summary of all Graces of the ville and Caius College, Cambridge. Senato, Degrees conferred during the year 1801, and University news generally are AN ELEMENTARY TREATISE ON given. Trilinear Co-Ordinates. 5. The Regulations for the LOCAL EXAMIxation of those who are not members of the The Method of Reciprocal Polars, and University, to be held this year, with the the Theory of Projectiles. names of the books on which the Examination will be bitnod, and the date on which the 154 pp. (1861). Cr. 8vo. cl. 68. 6d. Examination will be held. The object of the Author in writing on Published annually in January. this subject has mainly been to place it on a basis altogether independent of the ordinary Cartesian System, instead of regarding it is only a special form of abridged Nota- A short chapter on Determinants has Cambridge University been introduced. Examination Papers, Senate House Mathe- Crown 8vo. 184 pp. 2s. 6d. matical Problems. A Collection of all the Papers set at the With Solutions. Exantingtions for the Degrees, the various Triposes, and the Theological Certificates 1848-51., By FERRERS M JACKSON. 8vo. 158. 6d, in the University, with•List of Candidates Examined and of those Approved, and an 1848-51. (RIDERR). By JAMESON. 8vo. 73. 6d. 1854. By WALTON and MACKENZIE. 8vo. Index to the subjects. 1860-61. 108. 6d. Previous Examination, 1860—Previous 1857. By CAMPION and WALTON. 8vo. 88. 6d. Examination, 1861-B.A. Degree Examina- 1860. By ROUTH and WATSON. Crown 8vo. 78. 6d. tion, 1860-B.A. Degree •Examination, Jan. and May, 1861-Bachelor of Laws Examina The above books contain Problems and tion, 1860-Bachelor of Laws Examination, Examples which have been set in the Cam- 1861 Bachelor of Medicine Examinations, bridge Senate-houne Examinations at various 1861-Classical Tripos, 1861–Moral Sciences periods during the last twelve years, together Tripos, 1861- Natural Sciences Tripos, 1861 with Solutions of the same. The Solutions -Smith's Prizes, 1861-Chancellor's Medals are in all cases given by the Examiners for Legal and Classical Studies, 1861. themselves or under their sanction.

·. 10 CAMBRIDGE CLASS BOOKS By C. W. UNDERWOOD, MA. By J. HERBERT LATAAM, M.A. Vice-Principal of the Collegiate Institution, Civil Engineer. Liverpool TIIE CONSTRUCTION OF A Short Manual, of Wrought-Iron Bridges. Arithmetic. Embracing the P: ical Application Fcp. 8vo. 96 pp. (1860 imp.cloth. of the Principles of Mechanics to 28. 6d. Vrought-Iron Girder-Work. The object aimed at by the Compiler of "The great merit of this book is that it this Manual to bring before wnior deals with practice more than theory. All students so much of the Theory of Arith- the calculations in the book connected with metic as may be fairly expected of them, and the strength of girders are based upon their to present it in such a form that the study actual application which abounds in practiof the Science may become to some extent cal investigations into girder-work in all & mental training. It is rather a Grammar its bearings, and will be welcomed as one of of Arithmetic than a treating on that sub-ithe most caluadir contributions yrt made to ject, and should for the most part be como this important branch of engineering."- mitted, to memory. It will be found well ATHEN&UM. adapted for rica voce examination, and enable candidates to prepare themselves for the Local University Examinations. The By JONN E. B. MAIOR, Definitions are briefly and carefully worded. Fellow and Classical Lecturer of St. John's Each rule is stated so as to include the proof College, Carubridge.' of it where thl was possible. 1. Jyvenal. By H. A. MORGAN, N.A. Fellow of Jesus College, Cambridge. Trith English Notcs. A Collection of Mathe- 464 pp. (1854). Crown 8vo. cloth. 108. Od. matical Problems and "A School edition of Juvenal, which, for Examples. really ripe scholarship, extensive acquaintance with Latin literature, and familiar With Annivers. knowledge with Continental criticism, an- 190 pp. (1858). Crown 8vo. 68. 64. cient uod modern, is unsurpassed, we do not say among English School-books, but among • This book contalus a number of problems, English editions generally "-EDINBUROU chiefly elementary, in the Mathematical sub REVIEW. jects usually read at Cambridge. They have 2. Cicero's been selected from the papers set during late years at Jesus College. Very few of Second Philippic. them are to be met with in other collec. tions, and by far the larger number are due With English Noterets to some of the most distinguished Mathematicians in the University. 168 pp. (1861). Fcp. 8vo. cloth. 58. The Text is that of Halm's 2nd edition, Elementary Statics. (Leiprig, Weidmann, 1858), with some corrections from Madrig's 4th Edition (Copen- By the Rco. GEORGE RAW LINSON, hagen, 1858). Halm's Introduction has been rofessor of Applied Sciences, Elphinstone closely translated, with some additians. Ilis Coll., Bombay. notes have been curtailed, omitted, or cn- Edited by the Rep. E. STURGES. M.A. larged, at discretion; passages to which he gives a bare reference, are for the most part Rector of Kencott, Oxfordshire. printed at length; for the Greek extracts an A Crown 8vo. cloth. English version Kas been substituted. 48. 6d. large body of notes, cbiefly grammatical and historical, has been added from various This work is published under the au- A list of books useful to the thority of H. M. Secretary of State for India student of Cicero, a copious Argument, and for use in the Government Schools and an Index to the introduction and notes, com- Colleges in India. pleto the book, (160 pp.) 1860. sources, ·V. 1 FOR SCHOOLS AND COLLEGES. 21 By C. MERIVALE, B.D. By P. Frost, Jun., M.A. Author of “A History of Rome,' &c. Late Fellow of St. John's Cou. Cambridge. Sallust. Thucydides. Book VI. Tith English Notes: With English Notes, Map and Index. Second Edition. 172 pp. (1858). Fcap. cloth. 78. 6d. 8vo. 48. 6d. “This School edition of Sallust is precisely It has been attempted in this work to fawhat the School edition of Latin author cilitate the attainment of agcuracy in transought to be. No useless words are spent lation. With this end in vicw the '1 xt has in se, and no words that could be of use been treated grammatically. are spared. The text hag been carefully collated with the best editions. With the work is given a full current of extremely wellselected annotations."-THE EXAMINER. By WRIGIIT, M.A. The “ CATILINA” and “ JUGURTHA" may be had separately, price 28. 6d. each, bound Head Master of Sutton Coldfield School. rn cloth. Help to Latin Grammar. 'By B. DRAKE, M.A. Fith Easy Exercises, both English and Latin, Questions and Vocabulary. Lato Fellow of King's Coll. Cambridge. 1. Demosthenes on the 175 PP. (1855). Crown 8vo. cloth. 48. 6d. Crown. “This book aims at helping the learner Tith English Notes. to overstep the threshold difficulties of the Latin Grammar; and never was there Second Edition. To which is prefixed better aid offered alike to teacher and ÆscHINES AGAINST CTESIPHON. scholar in that arduous pass The style is With English Notes. at once familiar and strikingly simple and lucid; and the explanations precisely hit the 287 pp. (1860). Fcap. 8vo. cloth. 58. difficulties, and thoroughly explain them. It will also much facilitate the acquirement The first edition of the late Mr. Drake's of English Grammar."-ENGLIAH JOURNAL edition of Demosthenes do Corona having | OF EDUCATION. met with considerable acceptance in various Schools, and a new edition being called for, the Oration of Æschines against Ctesiphon, in accordance with the wishes of many teecum has been appended with useful 2. Hellenica. notes by a competent scholar. A FIRST GREEK READING BOOK. 2. Æschyli Eumenides. From Diodogus and Thucydides. With Vocabulary. With English Verse Translation, Copious Introduction, and Notes. Second Edition. 150 pp. (1851). Fcap. 8vo. 144. pp. (1853). 8vo. cloth. 38. 6d. "Mr. Drake's ability as a critical Scholar is known and admitted: In the edition of In the last twenty chapters of this volume, the Eumenides before us we meet with him Thucydides sketches the rise and progress also in the capacity of a Poet and Historicul of the Athenian Empire in so clear a style Essayist. The translation is flowing and and in such simple language, that the author melodious, elegant and scholarlike. The doubts whether any easier or more instruo. Greek Text is well printed : the notes are uve passages can be selected for the use of clear and useful."-GUARDIAN. the pupil who is commencing Groek, ·12 CAMBRIDGE. CLASS BOOKS By J. TRIGAH, M.A. O By EDWARD TRRING, M.A. Head Master Sutton Colfield School: IIed Master of Uppingham School. 3. The Seven Kings of 4. Elements of Grammar Rome. Taught in English. A First Latin Reading Rank, abridged With Questions. from Liey, by the omission aficult passages, tcith Notes and index. Third Edition. 136 pp. (1860). Demy 18mo. 28. Second Edition 138 pp. (1857). Feap. 8vo. cloth. 3e This work is intended to supply the pupil 2. The Child's English with an easy Construing-hook, which may, at the samne time, be made the vehicle for Grammar. instructing him in the rules grammar and principles of composition. These branches New Edition. 86 pp. (1869). Demy of the sidy of Latin seem to the author to 18mo. 18 have hitherto been kept too much apart. Bors have construed their Delectus, or Eu- The Author's effort in these thinks tropius, or epos, and have gone elsewhere has been to point out the broad, beaten, for their grammatical exercises. Sor can everyday path, curefully aroidiſk digres this be wondered at. An educated man sions into the byeways and eccentricities must feel positively ashamed of taking his of language. This work took its rise from pupils away from our good English authors, questionings in National Schools, and the and setting before him insteud a Delectus whole of the first part is merely the writing or Eutropine. He therefore skims over them as lightly, and escapes from thein as qeickly have been used already with success. out in order the answers to questions which The as possible, and has recourse for his com- study of Grammar in English has been much position lesson to one of the books which swarm from our educational noglected, par by some put on one side us an impossibility. There was perhaps much No. To remedy these evils this book has ground for this opinion, in the medley of been published. Here Livy tells his own arbitrary rules thrown before the student, pleasant stries in his own pleasant words. which applied indeed to a certain number of What is omitted, is that which no one can instances, but would not work at all in many wish a beginner to learn, and which may others, as must always be the case when be better learnt elsewhere. Let Livy be the principles are not put forward in a language merster to teach a boy Latin, not some English full of anbiguities. The present work does collector of sentences, and he will not be not, therefore, pretend to be a compendium found a dull one. of idioms, or a philological treatise, but a Graminar. Or in other words, its intention is to teach the learner how to speak and write 4. Vocabulary and Ex- correctly, and tº understand and explain the speech and writings of themIts ercises on “The Seven success, not only in National Schoork, from practical work in which it took its rise, but Kings of Rome." also in classical schools, is full of encouragement. 94 pp. (1857). Crown, 8vo. cloth. 28. 6d. 3. School Songs.. The Vocabulary is published apart from the Text in order to suit the views of those A COLLECTION OF SONGS FOR who prefer their pupils to consult a general SCHOOLS. dictionary, but it may also be had bound together with the “SEVEN KINGS OF Roxe,' it preferred. As the aim of the Text is to WITH THE XUBIC ARRANGED FOR FOUR teach the elements of grammar, so the Ex- VOICES, ercises are intended to test the Pupil's knowledge of grammar. Indeed there is hardly Edited by Rev. E. THRING und an ordinary Latin construction which is not H. RICCIUS. illustrated in the text, explained in the notes, and proved in the Exercises. Music Size. 78. Od. 1 ·23 FOR SCHOOLS AND COLLEGES. BY EDWARD TIRING, M.A. 2. Head Master of Uppingham School. Notes for Dectures ox Confir- 4. A First Latin Con- mation: With Suitable Prayers. struing Book: By C. J. VAUGHAN, D.D. Fourth Edition, 104 pp. (1855)., Fcap. 8vo. 28. 6d. Pp. (1862). Fcp. 8vo. 18.60 This Construing Book is drawn up on the same sort of graduated scale do the Author's This work, originally prepared for the use English Grammar. Passages out of the of Harrow School, is published in the belief best Latin Poets are graduully built up into that it may assist the labours of those whom their perfect shape. The few words altered, or inserted as the passages go on, are printed are engaged in preparing candidates for in Italics. It is hoped by this plan that the Confirmation, and who find it difficult to lay leurner, whilst acquiring the rudiments of their hand upon any one book of suitable language, may store his mind with good poetry and a good vocabulary. instruction at once sufficimtly full to furnish a synopsis of the subject, and sumciently elastic to give free scope to the individual judgment in the use of it. 1 Religious Class Books. 3. 1. The Church Catechism Illus- By C. J VAUGHAN, D.D. treted and Explained. Ву Jarl Master of Harrow School. ARTHUR RAMSAY, M.A. 18mo. St. Paul's Epistle to the cloth. Romans. 4. The Greek Text with English Notes. Hand-Book to Butlerts Ana- Second Edition. Crown 8vo. cloth. logy. By C. A. SWAINSON, M.A. (1861). 58. 55 pp. (1856). Crown 8vo. Is. 6d. By dedicating this work to his elder Pupils at llurron, the Author hopes that he 5. sufficiently indicates what is and what is not History of the Christian Church to be one for in it. Anis impression, derived from the experience during the First Three Cenof many years, that the Epistles of the New Testament, no less than the Gospels, are turies, and the Reformation capable of furnishing useful and solid inin England. By W. SIMPSON, struction to the highest classes of our Public Schools. If they are tauglit accurately, not M.A. Fourth Edition. Fcp. 8vo. controversially; positively, not negatively; authoritatively, yet not dogmatically; taught cloth. 38. 6d. with close and constant reference to their Literal meaning, to the connexion of their parts, to the sequence of their argument, as 6. well as to their moral and spiritual instruction ; they will interest, they will inform, Analysis of Paley's Evidences they will elevate; they will inspire a reveof Christianity. By CHARLES rence for Scripture never to be discarded, they will awaken a desire to drink more H. CROSSE, M.A. 115 pp. (1855). deeply of the Word of God, certain hereafter .18mo. 38. 6d. to be gratified and fulfilled.

·CAMBRIDGÈ MANUALS FOR THEOLOGICAL STUDENTS. UNIKRULY PRINTED AND BOUND. Bs А This Series of THEOLOGICAL MANUALS has been publishedc with the aim of supplying Books comcise, comprehensive, and accurate; convenient for the Stufient, and yet interesting to the general reader. 1. History of the Christian able for the desired object. In the course Church during the Middle of the last twenty years the whole question Ages. By ARCHDEACON HARD- with great learning and avcurate research, WIEK. Second Edition. 482 pp. and it is mainly with the viow of epitomizing (1861). With Maps. Crown 8vo. their extensive publications, and correcting cloth. Yos. 6d. by their help the errors and miscountans which had obtained currency, that the pro- This Volamo claims to be rogarded as an bent volume has been put together. intogral and independent treatise on the Hodieval Church. The History common ces 4. History of the Canon of the with the time of Gregory the Groat, because New Testament during the it is admitted on all hands that his ponti First Four Centuries. Aicato hocamo a turning point, not only in the fortunes of the Western tribes and BROOKE FOSS WESTCOTT, pations, but of Christendom at largo. M.A. 594 pp. (1855). Crown 8ro. kindrod reason has suggested the proprioty cloth. 128, 6d. of paquing at the year 1520,--the year when Luther, having been extruded from those The Anthor has endeavourod to connect Churches that adhered to the Communion the history of the New Testament Canon of the Popo, established a provisional form with the growth anol consolidation of the of government and opened a fresh era in Church, and to point out the relation the history of Europe. existing between the amount of evidence for the authenticity of its component parts 2. History & the Christian and the whole mass of Christian literature. Churcł during the Reforma- Such a method of inquiry will conrey both the truest notion of the connexion of the tion. By AKCHDEACON HARD- written Word with the livmg Budy of Christ, WICK 459 pp. (1856). Crown and the surest conviction of its divine auk 8vo. cloth. 108. 6d. thority. This Work forms a Soquel to the Author's 0. Introduction to the soudy Book on The Middle Ages. The Author's of the GOSPELS. By BROOKE wish has been to give the roador a trust- FOSS WESTCOTT, M.A. 458 pp. wortby version of those stirring incidents which mark the Reformation period. (1860). Crown 8vo. cloth. 1 s. 6d. The title of this book will oxplain the 3. History of the Book of Com-chief aim which tho Author had in view. It mon Prayer. With a Kationale is intendod to be an Intruduction to the of its Offices. By FRANCIS Study of the Gospels. The Author has thoroforo confined himself in many cases to the PROCTER, M.A. Fifth Edition. more indioation of linor of thought and 464 pp: (1860). Crown 8vo. cloth. inquiry from the conviction that iruthi 108. od. felt to be more precious in proportion as it is opened to us by our own work. In a subject The Subject of Cia Book has beon already bvhich involves s Vast a literature much treated by armorous writers of distinction. must have beon overlooked; but the Author When the prosent series of Manuals was has made it a point at loast to study the reprojected, it did not appear that any one of searches of the great writers, and consciously the existing volumes taken singly was avail. kto nøglest none. 1 ·15• FORTHCOMING BOOKS: An Elementary Treatise on the Planetary Theory. By C. H. A. CHEYNE, B. A., Scholar ofrot. John's College. Crowa 8vo. cloth. 68. 6d.g [Just ready. 2. A' Treatise on Geometry of Three Dimensions. By PERCIVAL FROST, M.A., St. John's College, and JOSEPH WOLSTENHOLME, M.A., Christ's College, Laulbridge. 8vo. 188. The First Portion has been issued for the convenience of Cambridge Students : the concluding portion will appear before Christmas, 1862. 3. An Elementary Treatise on Natural Philosaphy. By WILLIAX THOMSON, LL.D., F.R.S., late Fellow of St. Peter's College, Cambridge, Professor of Natural Philosophy in the University of Glasgow; and PETER GUTHRIE TAIT, M. A., late Fellow of St. Peter's College, Cambridge, Professor of Natural Philosophy in the University of Edinburgh. With numerous Illustrations. [Preparing 4. An Elementary Treatise on Quaternions. With numerous Examples. By P. G. TAIT, M.A., Professor of Natural Philosophy in the University of Edinburgh. Preparing. 5. Ficeto Book of Atgebra. For Schools. By J. C. W. ELLIS, M.A., and P. M. CLARKE, M.A., Sidney Sussex College, Cambridge. [Preparing. 6. Aristotelis de 'Rhetorica. With Votes and Introduction. By E. M. COPE, M.A., Fellow and Assistant Tutor of Trinity College, Cambridge. 7. The New Testament in the Original Gredak. Text revised by B. I WESTCOTT, M.A., and F. J. HORT, M.A., formerly Fellows of Trinity College.

·16 MACMILLANLAND CO'S CAMBRIDGE SCHOOL •CLASS BOOKS. I. Euclid for Colleges and Schools. By I. TODHUNTER, M.A., F.R.S., Fellow and Principal Mathematical Lecturer ofe$t. John's College, Cambridge. 18mo. 38. 6d. An Elementary Jethtin Grammar. By H. J. ROBY, M.A., Under Master of Dulwich College Upper. whool, late Fellow and Classical Lecturer of St. John's College, Cambridge. 18mo. 28, 6ch An Elementary History of the Book of Common Prayer. By FRANCIS PROCTER, M.A., Vicar of Witton, Norfolk, late Fellow of St. Catharine's College, Cambridge. 18mo. 26. 6d. There volumes, forming the first portion of Macmillan and Co.'s ELE- XENTARY SCHOOL CLASS BOOKo, are handsomely printed and bound. All the volumes of the series will be published at a low price to ensure an extensive sale in the Schools of the United Kingdom and the Colonies. po The following will shortly be published: The School Class Book of Arithmetic. By BARNARD SMITH, M.A., late Fellow of St. Peter's College, Cambridge Algebra for Beginners. By L. TODHUNTER, M.A., F.R.S. The Bible Word Book. A Glossary, of old. Enigla Bible Words with Illustrations. By J. EASTWOOD, WA, St. John's Bollege, Cambridge, and lpcumbent of Hope," near Sheffield, and W. ALDIS WRIGHT, M.A., Triņity College, Cambridge, Other Volumes will be annonwood, in due course. Printed by ouma, Sidney: Street, Cambridge.

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