A treatise of algebra, in three parts. Containing

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of - , R is quarto the excessan ita ste sum. of the reciprocais of the segment termina ted by the polii mand the curve surpasies the sum of the reciproces of the segment terminate by the fame mini and the tangents ΑΚ, Bia c. aut as osten cinis exces comes out negative, the chord DR is maestae, on the other Me of the piant and the rule above de scribe is alway to ae applied for distinguishing the Mns os the irems. Is in right linem bisem theangle EDT, made by the right linei and the eangent In the theoremaecomes a litue more simple.

I6. Fro the iam principi Allows a generaltheorema whio the variationis curvatur is deter- naiqed, o the measure of the anse of contae contain

by the curve i in the osculator circle, in an geometrica line; et a bries explicatio of the variation os curvature mustae premis , since this is no clearly describe is authors. ver curve is snt fio iis tangen by iis curvature, os,hic the measere is in sanie as of the angle o contin containe by the curve

an tangent; and in litis manne a curve is bent homit ciculator circle hy the variationis it curvature, o inllic variatio the meatur is the fame as of ille angier os contae contained by the curve an osculatorycircle. et the right line E perpendicular to the Fig. tangenti meet the curve in Mand the osculatorycircle in , and the variation os curvatur Will e

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may be collecte stom Art. 9. of the reati se of Fluxionc in generat the variation os curvature,illae

ultimatet a GD es a circle ' determining the curvature os other figur ei; ut in me ure the variation os curvature, which is nothiniin, circle, a par bola or seme coni lection icto be applied No as osilie circle indefinite in number hic ma Duch agi en curve in a give potnt, ne oni is calle ossi Ialory, hic so losely ouches the curve that o other cana drawn et ween his and the curvea in lihemaniter of ali parabolas,hich have the same curvature wit the litie prcipo sed at a gi ven o in t so these a realis infinite in number that ni has the fame variation os curvature, hic hiso oni to uches the arc of

Lot D tabe the arctos a curve, DT a tangent, TE , a right in perpendicular to the tangent, and let therectangle ET 'Κ be always equa to the square of the tangunt DT, and in curve SKF ille locus es me potnt Κ, whici meet the lineis perpendicula toti, curve in f, and which touches ille right linem V , in S cuuing the lausent Dan The righi lineis wil be the diamete os ille osculator, circle, and Dy iugiis die iura j v illis the centre os curvature

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GEOMETRICA LINES. 453now initoined is in and SDN be made qualto the anglei vinon the offer de of the right line DS, and the right line V meet ille osculator circlela, ohen the parabola described with the diameterandia rameter DN, and whic to uches the right line DT 1 D, in ill e that wliola contae with the line proposed in D,ill e the losest an most persei orniearest that ca be described. ut allisther parabolas, det ibed wit an other chor of the osculato circle although describe with. the diameter an parameter, and ouching the right line in D, have the fame curva

a the radius , and the variation es curvatur as

hic is the measure of the an te o contin containes Meen the curve and the ostulator circle. Now of thes the oneris asil derive seom the other.

The variationis the radius O curvatur m an Curve

D is a the tangent of the angi DUMO DU, and . 'in an parabolarit is alway as he tangent of the ingle contai hed by a diameter passi nil luough the poliat os contae an a right in perpendicular to the curve. Thes thing ma be deduce stom the solioWing generat theorem.

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right anglesie bisected by the right linea ob, hichie meet the proposed geometrica line in the poli 0 D, hinc anxiet the tangent ari , dcci, heidra n, cui the right linei in the minis Ma M. the by

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sure of the angi of contain me eontaine by the curve alui osculatoryclaese, o the variationi curvature, is as and theres etas Te - , Mi 8. No the vallat tot of the radius os curvatur o the qualit os it describe by Newton, is most easilycolleeted sto the former For SI, SK SL,inc beingjoined, iliis variatio of the osculator radius,illae as the exces hic therium of the tangent of the

curvature rom intoward eris diminished, and the radius of the osculator circle is increased, as osten as ille M De of the curve ouehes the circular arc externalty oriastes between the circle an tangent, there-

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more geometrica sorm sem lemmas area be pre-

inised by whic the doctrine of the harmonica division es right lines is made more fuit ani generat. In

lines are, by Dei Hira calle Harmonicali sui

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monicali by ou right lines rawn fiam the lamepoint, that an right line wllic meet these laures mes wil aliti e ut harmonicali by the same; ut thatthat hic licis parallel to ne of the four is divided into equa segments by the re maining three . . Leti be