APPENDIX: BEING A TREATISE CONCERNING THE GENERAL PROPERTIES OF GEOMETRICAL LINES. TRANSLATED FROM THE LATIN BY JOHN LAWSON, B. D. Ff CONCERNING THE GENERAL PROPERTIES OF GEOMETRICAL LINES. NONCERNING the lines of the fecond order, or the CONC conic fections, the ancient and modern geometers have written very fully; concerning the figures which are referred to the fuperior orders of lines, little has been delivered before NEWTON. That most illuftrious man, in his tract concerning the Enumeration of Lines of the Third Order, has revived this fubject, which had long lain neglected, and has shewn it to be worthy of the geometer's notice. For the general properties of these lines, which he has laid down, are fo confonant to the known properties of the conic fections, that they seem to be conformable to the fame law, and from his example many others have been fince induced to make this fubject their ftudy, and have clearly comprehended and explained the analogy which there is between figures of fuch very different kinds. The pains which they have been at in the illuftration and further inveftigation of these matters, have deservedly met with applaufe, fince there is nothing in pure mathematics which can be called more beautiful, or that is more apt to delight a mind defirous of inveftigating truth, than the agreement and harmony of different things, and the admirable connection of the fucceeding with the preceding, where the more fimple always open the way to thofe which are more difficult. Moft of the general properties of lines of the third order, delivered by Newton, relate to fegments of parallels and afymptotes. Some other of their affections, of a different kind, I have briefly pointed out in my Treatife of Fluxions, lately published, Art. 324, and 401. The famous Cotes formerly difcovered a moft beautiful property of geometrical lines, hitherto unpublished, which has been communicated to me by the Rev. Dr. Robert Smith, mafter of Trinity College, Cambridge, a gentleman not lefs remarkable for his learning and works, than for his fidelity and regard for his friends. Whilft I had thefe under confideration, fome other general theorems offered themselves; which, as they feem to conduce to the augmentation and illuftration of this difficult part of geometry, I have thought fit to throw together, and briefly to expound in order, and demonstrate. SECTION I. Of Geometrical Lines in general. § 1. LINES of the fecond order are defined by the fection of a geometrical folid, viz. a cone, whence their properties are beft derived by common geometry. But the nature of the figures which are 2 referred referred to the fuperior orders of lines is different. To define and draw out their properties, general equations must be applied, expreffing the relation of the coordinates. Let x reprefent the abfciffa AP, y the or- Fig. 1. dinate PM of the figure FMH, and let a, b, c, d, e, &c. denote any invariable coefficients; and having the angle APM given, if the relation of the co-ordinates x and y be defined by an equation which, befides the coordinates themselves, involves only invariable coefficients, the line FMH is called a geometrical one; which indeed by fome authors is called an algebraical line, by others a rational line. But the order of the line depends upon the higheft index of x or y in the terms of the equation freed from fractions and furds, or upon the fum of the indices of both in a term where that fum is the greateft. For the terms x2, xy, y2 are equally referred to the second order; the terms x3, x2y, xy, y3 to the third. Therefore the equation y ax + b, or y — ax―bo, is of the firft order and denotes a line or the locus of the first order, which indeed is always a right line. For let there be taken in the ordinate Fig. 2. PM the right line PN, so that PN be to AP as + a to unity; let AD, parallel to PM, be made equal to + b, and DM, drawn parallel to AN, will be the locus to which the proposed equation will anfwer. For PM-PN + NM (a x AP + AD) ax + b. But if the equation be of the form y = ax b, or y == ax + b, the right line AD, or PN, is to be taken on the other fide of the abfciffa AP; for the contrary fituation of right lines anfwers to the contrary figns of the coefficients. If the affirmative values of x denote right lines drawn from A, the beginning of the abfciffa, to the right hand, the negative values will denote right lines drawn from the fame beginning to the left; and in like manner if Ff3 the |