A treatise of algebra, in three parts. Containing

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ne potnt Will meet the curve in the fame piant.

No the potnti is determined by dra in the line PC, hic touches the curve in an cut it in Q; forci AC MJoined, it,ill meet again the curve in ine in B. 1 o since the potnis F, Κ, P, are in the fame right

the fame right line, and Do each of them e drarunso ur lines ouch in the curve in laur of her po in is, arieth line drawn through an lino potnts of contac , illat way cut the curve in sume ther poliat o contact;

and our of thus lines ill alWaysias inroughabet me polim os contacti

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other and the curve in sonae potiri of the curve

B, and the tangent a the minis Wand 'illmeet in a mint of the curve C so that thethree Dint A, may be in ille fame right line.

For let the tangent at the pnin P e drau n, hic lite meet the curve in C, arid te AC cut the fame in B and ΒΚ, Laein drainn, ill e tangent a me potnis mand L, by the preceding. et the line I,Vmeet the curve in and f the in GK does notpas through in let i meet the curve in There-

fore, ecause in three minis , , Q; are in thesime right ine, ut the tangent at L and F cut in Bandra, it sollows by Prop. VII. that the tangent atine min q,illias also throuo the potat C. There-sore both lines C , much the curve, the sirst in

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taken in a line of the hird order so that tangent attheloinrei and G meet in ome potnt of the curve, and tangent a theloint Κ andi meet also in lamepol ni o the curve F and GL drawn Wil meet in apoint of the curve, an FL and G dram,ill meet eae oster in a piant of the curve.

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isitum, and so the right line INm is give in potation and in lita manno the potat Lis at a line gruenin position. v. a. v89 Corol. The politis G andra coincidire, heptant m ill also coincide illi the piant G. There-- fore let I oine and n et Isinini, an CF drawn and meet the line III in Ε, in sine DEJoinedwill be ille locus os ille potes K, Wherem and FP

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quadrilatera inscribe in a figure, hos sigangle to uch a line of the thir order, as in

Prop. XVI. et line IC, CH, HI rawntouehing the curve in three Iminis in P L, Whicli aremo in the fame right lines le IGjoined meet the tangent CH in D, and Hymeet the tangent C I in E ali potnis D, Κ, Ε, illbe in the fune right line, hicli Will ouch the me in the potnt K.

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tangenis drawn rom each of these to the curve Minggive in position the si potnts of contae are determine hy this proposition. te A, B, C e the threerive potnts of the curve in ne right line, AM and A tangetit fro A, BMI and BDE tangent stom B, whic meet the forme in M, and i an let

ini, and lastlyne the tangent Bh in No thoughthe problem in his case is determinate, et it admits of severTil solutions. For different lines of the third Order, hut definite in number, may be rawn throughthe three potnis A, B, C, ouching the si right lines

given in possitio AM, AN BM, BD, CD, and Ε.For let Ne meet the tangent CD in Amid the tangentC in q, I the tangentis in I, ΕΗ the tatuent Amin ranc ille tangent M in i, and m, the tangetit BD in L and a line of the inita orde Whic sitiines

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peus that the problem does notaecome impossibie be- cause i is necessar that three right lines LV Ald, ixd

95. PROP. XIX. a.e D, E, F be potnis d. s. in adine of the thir orde in the fame right sine, an let there e three lines ouching the curve in theseioint parallelao ach other. In