Cales. 6 B, C and AB, BC the fide R one of themi C. AB, BC, and C one of R B, two fides the other 7 and the included angle. angles. Solution. cos. B:: tan. AB: tan. BD, (cafe 2.), and tan. C: tan. B:: fin. BD: fin. DC; and DC is less than DB, if B and C be of diff. affection; or lefs than the fupplement of DB, if B and C be of the fame affection. In other cafes, DC is ambiguous. If B and C be of the fame affection, BC is the fum of BD, DC; otherwife it is their difference. cos. B tan. AB: tan. BD, (cafe 2.), and the difference of BC and BD is DC. And fin. DC: fin. DB: tan. B: tan. C, (p. 3.), and B, C are of the fame affection, if BC be greater than BD; otherwife they are of diff. affection. AB, BC, and AC the third Find BD and DC as in the laft B, two fides fide. 8 and the in cluded angle. cafe, then cos. BD: cos. DC:: A,B, andAB, C the third R: cos. AB :: tan. B: cot. BAD, (cafe 3.), and the diff. of BAC, A, B, and AC one of Find BAD and DAC, as in the AB, two the other angles and fides. the inclu ded fide. laft cafe; then cos. DAC :{ Spb. Trig. Sph. Trig. Cafes. Given. Sought. Solution. AB, AC, BCB one of the Let the perp. AD fall within, 12 fides. or be the nearest to B or C that falls without, tan. BC : tan.fum of BA, AC :: tan. diff. of BA, AC : tan. E, and E added to BC, gives the fegment nearest the greater fide, if the fum of AB, AC be lefs than 180°; otherwise it gives the fegment neareft the lefs fide. (Prop. 22.). And tan. AB: tan. BD : R : (cafe 12.). Otherwife, Let D be of AB, BC; then : : cos. B. the diff. the rect. fin. AB, fin. BC rect. fin. fum and diff. of D, and AC:: R2: fin 2.4B. (P. 23). Otherwife, Let P be the pere. meter; then rect. fin. AB, fin. BC rect. fin. P. fin. diff. of P, AC:: R2: cos 2, B. (24). A, B and CAC one of With the supplement of either the three the fides. angles. of the angles A, C, and the measures of the other two angles, suppose a triangle made; and in it find the angle oppofite to the fide which is the measure of the angle at B, and the measure of the angle thus found is AC. NOTES. NOT E S. BOOK I. TH DEFINITIONS. THE ift definition wants a condition to make it complete, Book I. for to have no magnitude is not peculiar to a point: This condition is now inferted from Dr Hooke, who fays, that a point has pofition, and a relation to magnitude, but has itself no magnitude: It may alfo be faid to be an indivisible mark in magnitude, as Tacquet has it: Or, it may be faid to be a fign ufed for determining pofition and the extremities of lines, for the name onto appears to have been given to it from its ufe. The 8th definition is left out, because it does not belong to the Elements; nor can it be explained, fo as to be understood by beginners, as is obferved by Dr Simfon. The 13th definition is alfo omitted, because it is useless in a tranflation, its only defign being to explain a Greek word. And the 19th, which is the definition of a fegment, is left out here, because it is given in the third book, which is its proper place. And the definition of the radius of a circle is introduced, because it is very frequently used by Geometers. These are all the alterations that have been made in the definitions of this book; but many more might have been made with propriety. The first nine definitions might have been given in the form of an introduction, for they are none of them geometrical, except the feventh, as amended by Dr Simfon. The terms by which a line and a fuperficies are defined, give fome explanation of the meaning of these words, but give no geometrical criteria by which to know them; and the best way of acquiring proper ideas of them, is by confidering their relation to a folid, and to one another, as Dr Simfon has done. BOOK I. "It is neceffary to confider a folid, that is, a magnitude which has length, breadth, and thicknefs, in order to understand aright the definitions of a point, line, and fuperficies; for all these arife from a folid, and exift in it: The boundary, or boundaries which contain a folid, are called fuperficies, or the boundary which is common to two folids, which are contiguous, or which divides one folid into two contiguous parts, is called a fuperficies: Thus, if BCGF, be one of the boundaries which contain the folid ABCDEFGH, or which is the common boundary of this folid, and the folid BKLCFNMG, and is therefore in the one as well as the other folid, it is called a fuperficies, and has no thickness: For if it has any, this thicknefs muft either be a part of the thickness of the solid AG, or the folid BM, or a part of the thicknefs of each of them. It cannot be a part of the thickness of the folid BM; becaufe, if this folid be removed from the folid AG, the fuperficies BCCF, the boundary of the folid AG remains ftill the fame as it was. Nor can it be a part of the thickness of the folid AG; becaufe, if this be removed from the folid BM, the fuperficies BCGF, the boundary of the folid BM, does nevertheless remain; therefore the fuperficies BCGF has no thicknefs, but only length and breadth. The boundary of a fuperficies is called : Ε G M F N D a line; or a line is the common boun- The boundary of a line is called a point, or a point is the common boundary or extremity of two lines that are contiguous: Thus, Thus, if B be the extremity of the line AB, or the common Book I. extremity of the two lines AB, BK, this extremity is called a point, and has no length: For, if it has any, this length must either be part of the length of the line AB, or of the line BK. It is not part of the length of the line KB; for, if the line KB be removed from AB, the point B, which is the extremity of the line AB, remains the fame as it was: Nor is it part of the length of the line AB; for, if AB be removed from the line KB, the point B, which is the extremity of the line KB, does nevertheless remain; therefore the point B has no length: and because a point is in a line, and a line has neither breadth nor thickness; therefore a point has no length, breadth, nor thickness. And in this manner, the definitions of a point, line, and superficies, are to be understood." No definition of a straight line has been given that is unexceptionable, though many of the ancients attempted it, as Proctus obferves, who has allo preferved their definitions. That given in the Elements, viz. its lying evenly, equally, or uniformly between its extremities, expreffes the nature of a straight line too metaphyfically: its meaning is, that a straight line has not a convex and a concave fide; but that both fides are alike. But the distinguishing character of a straight line, according to Euclid, is, that it is impoffible to apply one part of it to another, or one ftraight line to another, without their coinciding. All other lines require fome artifice in applying them to one another, in order to make them coincide; but no fuch artifice is neceffary in the cafe of straight lines, for they always coincide in whatever way we proceed to apply them to one another. That this was Euclid's idea of a straight line, is manifest from the fourth and eighth propofitions of the firft book, in which he does not fhew how the fides of the triangles are to be applied to one another, fo as to coincide, but takes it for granted, that AB in the fourth shall lie along DE, and that BC in the eighth fhall lie along EF, as foon as they are applied to one another. Now, this facility of application follows immediately from the uniformity of the fides of a straight line, and the other properties of a straight line are eafily deduced from it. Plato's definition, that the extremity of a straight line cafts a fhadow along the whole line; and that of Archimedes, that a ftraight line is the least of all the lines which have the fame extremities with it, were evidently defigned for particular purposes, and are not fit for the Elements. to two. The other definitions mentioned by Proctus, may be reduced One of them is, that a straight line is that of which the position is determined by the pofition of any two points of it. This is a property of ftraight lines which is fuppofed in the firft and |