A treatise of algebra, in three parts. Containing

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47 GENERA PROPERTIE or

drawn VII muching the osculator circle in H; et ΗDae joi neci, alui ilice the angle RDH is the complement of the angi Dr to a right one, DH,illzet Ur et ED J anxio the variation of the radius os curvature illae a the tangent of the angi RDII; and the right lines DR and DH coinciding that varia . tion vanishes.

neither arren nor unpleasant, nc besides the properties of these figures formeli deli vered by Λαυtun, there are mannother no vnwOrthy the attention fgeometers. I have hewn above, that a right line maycut a line of the hir orde in three potnis, ecausethere are three root of a cubi equation, hic may

aliae res. No a right line hicli cur a line of thethir orde in lino politis, necessarii meet the fame in semesinita Dint, or is parallel to the asymptote of the

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GEOMETRICAL NE . 475. the curve, in whicli case i is selyto meet it at an im 'finite distance: for i two oot os a cubi equationare reat, the hir Wil necellarii he real Hencelaright line hic to uches a line of the thir order, al-ways ut it in s me potnt, since the contactris to belooked pon a tino coincident intersections. ut a right line whicli ouches the curve in theloin os contrar flexure, is a me fame time to e stremed a secant. Whe two res of the curve meet hach othe sistere is a dotiis mini omeri an a right line whicli -Duchesinister arc there in the iam piant ut theother. ut an other right line diawn sto the doublepotes ut in curve in ope ther potat ut no in

more.

6 2. PROP. I. et there e t o paralleis, each of whicli let cui a line of the third orderin three potnis a right line whicli se cut bothos the paralleis that the sum of the two pares of the parallel terminatexat the curve o oneside of the cutim line may be qua to thethir par of the fame terminated at the curveo the ther sde of the cotting line, ill in illi maniae cut ait ther right lines parallelto these hici meet the curve in threeioint ;b Art. q.

6 53 PROP. II. et a right line give in

positio meet a line of the thii forde in threepoinis; et an two paralleis e dra n bothos hicli let cui the curve in s many oinis; and the solid contained unde the segments of

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the parallel terminate by the curve and the Iine give in position, illi in lae fame ratio. the solid unde the segments of this right sine terminared by the parallela by Art. s. These go propertie were form*rly exbibite is

Me 1, 6 PROP. III. The est remaining achithe precessire positi , let the right sine giveni positio meet a line of the thir orde inone oni potnt A, and the solici containexunder the segments PM, m Ρμ of one parallel ill alway be to the soli unde the segment pN,

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cris in Art. 6 and the harmonica mea he- tWeen the three lines Κ, L, Μ, ill beto the harmonica mean belween the two right

linea mandri P in the ratio os a to et by

646. PROP. V. et the right line PD re- a ut the pol P., let M e alwaystaen in D eques to the harmonical mea be-t Neen the three line PD, E, and F, and the locus of the win Μ willies right line, by Art. 28. An illis is a properi os these lines invente D

Due'. 'Rop. VI. Let there M three pinnis Fig. os a line of the thir Loide in the fame right line ciet right lines touching the curve in thesepoinis e drawn Which may cut the sam in three ther minis these three potnis,ill also hem a right line.

Let the right Ilinea GH meet a line of the hird or- de in the oinis and H. et the lines A, GB, C to uching the curve in these potnis ut thesime in the oinis A, B, C and these potnis,illae in a right line. For let AB hecioined, and this,ill

curvo

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. 73 GENERAL PROPERTIES OF

curve in an other pollit d. the tangent C in and the right line FGH in ' and since F

PM4,hicli cannotae, utilis the potiris', M and Ccoincide Therelare the right line AB passe through C. 58. Coro Henc is A, B, C be three pol nisi a line of the third orde in the fame right line, and APand BGae inglara est ouch the curve in F and G, and FG, ein joined, ut the curve again in H, H b ing oined, ill touch the curve in I . For is a right line mould ouch the curve in II, hic mouid noteut it in C ut in sonie other pollit, his mint,ouldbe with in three offera A, in the sun right lines teli,ould theresere cui a line os the stird orderin sis potnis. But this cannot be . Hr hi uponthis proposition in a disserent an ut telis expeditious, by deducing the same rom Prop. II. In like mannenis the right lineras also ouches the curve in I. an afheria drainn, meet the curve in Cit, eing oined, Willis a tangent at the o in Andris stoin thepolitis A, B, C, o a line os in thir orde silualedin theriam right line, e drawn a many right lines Ducbing the curve a canae drawn there Will a Vsbe three potare os contai in the fame right line.

36. 49. PROP. VII. . Fro anyioin os a lineos the thir orde letie drawn two lines touch-ing the curve, and et the lineboining the potnis of contaei ut the curve iii another poliat, thetangent to the curve at this ther Din and

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the curve.

Frem the piant A leti dram lines chiching the curve n F and let FG, eing oined, ut thecurve in II, and lectouch inessime in hi the line C

m,hence te AF and AG hein re drawn ouch the 'curve in F and ἰ, and la FG, eing oined, ut thecurve in , and AHaeing rawn will ouch the curve in H. For is the tangent at the polia H mould meet the curve in an other potia different from A the right line dia narum ibi potnt of meetincto inui tanti contra

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in the fame right line. For let F, G, H, e in thesame right line, stom hic tangent belu dram, marmeet in the same Dint of the curve a Which is no theptant os contrary flexures; et a be drawn ouching the Curve III a, and whic meet it in e and II, e ingjoined, ill ouch the curve in , by this proposition and so the lines an ab ,ould touin the curve in

the fame potat iri. Whicli is absurd.

3 3. PROP. VIII. Frum an potnios a late

of the thir order et e rawn three ines touch in the curve in three potnis; et a right lineboining tw of the potnts of contae meet the corve again, an a right line drawn stominat oint of meetincto the thir pinni fcontact ill again cut the curve in a poliat where a right line to the firs potat will ouchthe curve.

betara n hie lines AF, G, uici As, ouching the curve in the three poliat F, G, and let the line in Whicli uini t w of them, meet the curve againin N, and a line dra ista frona his poliat to the third

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Q and siticeo minus, and fare in the fame right litae, and the tangent a the poliat G and y isse inmugh A, it sollows by Prop. VII. that the tan gent at the polim N passes through C. An since thopolitis F, , ore in a right line, hut in tangenu F and Noemeet the curve in A and C, and AC is a

ine fame poliat of the curve touchinia line of the thirdorde besides that whicli ouches it in that fame poliat. For is sines miotae drawn stomaho semetiant of the curve ouchinii five pianis, more lines indefinite in number might be Mawn stom the sime polii Ductingste curve; ascis easti gatheredimis at oesi ore. No this Corollar we mali astet inard detrionstrate more eastir See belo , Art. 7.

6s PROP. IX. te three tangent to Fig. 38. the curve e ram rom a polat os contrant flexure, and a right ight line Joining the potnts of