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438 EmeRSON’j Arithmetic of Infir'tes, c.alter feveral manners, which will generate feveral forts of curves. Thus a circle is.defcribed by a line revolving round a fixed center; an ellipfis is defcribed by a flexible line moving round two fixed centers ; a cycioid is defcribed by a point in the circumference of a circle, whilft it turns round along a right line ; and fo of others.’We fball conclude this article with our author’s reafonmg prop. xvii. to prove, that 4 the properties of curves of a fuperior order, agree like wife with thofe in inferior orders.‘ An equation of a curve of the third order is denoted thus,y' 4. ax^-l Xyy-f- cxx-\--dx-\~e'Xy ■\-gx‘l-j-hx-\-kz=:Q. If fy £■ by be fuppofed =0, then the equation will becomejr-j- ax-\-b Xy-j- + dx-\- e~ °, which is one of a lower order. Now the properties of the curve belonging to the firft equation, muft h Id good of the curve belonging to the fecond eonation, with all the quantities that remain; and therefore its properties are included in the former. For all the difference is, th-ic fome line or lines become o and vanifh, and others become infinite, f me coincide, others become equal; likewiffe- fome points coincide, and others are removed to an infinite diftance ; yet, under thetc circumflantes, the general properties ftill hold good with the remaining quantities : fo that whatever is demon-ftrated generally of any order, holds true of the inferior orders. And, on the contrary, there is hardly any property of the inferior orders, but there is (bine fimilar to it in the fuperior ones.4 For, as In the conic fedtions, if two parallel lines are drawn, terminating at the fc£Hon ; the right line that bifleets thefe, will biffedfc all other parallels thereto ; and is therefore called the diameter of the figure, and the biflecfed lines ordinates » the in-teriVdtmn thereof with the cm ve, the vertex, and the interfec-tion of all ihe diameters the center; and that diameter, the axis* which is perpendicular to the ordinates. So likewife in higher curves, if two parallel lines are drawn, cutting the cutve in a proper number of points; the right line that cuts rhefe parallels fo, that the fuin of the parts cm one fide the line to the curve, be equal to ihr fum of the parts on the other fide, it will cut all other parallels in the fame manner, which cut the curve in as many points; then thefe parts may be called ordinates ; and the line to cuttin- them the diameter; the interfedtion of the diameter and curve, the 've*tex\ the in ter feel ion of two diameters the center-, the diamet-r perpendicular to the ordinates, if there be any, the axis. And when all the diameters concur in one point, that ts the ye.w a; center.
Newspaper Details
Monthly Review
London, Middlesex, GB
Tue, Dec 01, 1767
Page 19
Full Page
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USA 26 Jun 2024
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