Cales. Give!). Sought. Solution. Spo. Tris. B, C and AB, BC the fideR: cos. B :: tan. AB:tan. BD, two angles between the (cale 2.), and tan. C:tan. B:: 6 and the side angles. fin. BD : fin. DC; and DC is less than DB, if B and C be of diff. affection; or leís than the supplernent of DB, if B wise it is their difference. B, two sides the other (case 2.), and the difference of 7 and the angles. BC and BD is DC. Andi included sin. DC : fin. DB : tan. B : tan. G, (p. 3.), and B, C are wise they are of diff. affection. case, then cos. BD:cos. DC:: 8 and the in cos. BA : cos. AC (p. 3). In cluded BD, DC be of the same af- fection, BA, AC are of the ; otherwisel they are of different affection. A,B, and AB, C the third R:cos. AB::tan. B:cot. BAD, two angles angle. (case 3.), and the diff. of BAC, 9 and the in BAD is DAC, then sin. BAD : fin. DAC:: cos. B: fection. DAC : AC. If DAC, B be of the same affection, AC is less than than 90°. Cases, 10 Spb. Trig. Cafes. II : Given. Solution. or be the nearest to B or C fides. that falls without, tan. BC: tan. Ź sum of BA, AC :: tan. diff. of BA, AC : tan. | E, and IE added to BC, gives the segment nearest the greater fide, if the sum of AB, AC be less than 180°; otherwise it gives the seg. ment nearest the less side. (Prop. 22.). And tan. AB : tan. BD : Ri cos. B. (case 12.) Otherwise, Let D be the diff. of AB, BC; then the rect. fin. AB, fin. BC : rect. fin. sum and diff. of D, and AC:: R2:: fin 2. B.(P.23). Otherwise, Let P be the pere. meter; then rect. fin. AB, fin. BC: rect. fin. P. fin. diff. of P, AC:: R2:cosa, B.(24). A, B and CAC one of With the supplement of either the three the sides. of the angles A, C, and the measures of the other two angles, suppose a triangle made ; and in it find the angle opposite to the side which is the measure of the angle at B, and the measure of the angle thus found is AC. 12 angles. NOTES N O T E S. BOOK. !. DEFINITIONS. HE ist 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 inserted from Dr Hooke, who says, that a point has position, and a relation to magnitude, but has itself no magnitude: It may also be said to be an indivisible mark in magnitude, as Tacquet has it: Or, it may be said to be a fign ufed for determining position and the extremities of lines, for the name onestoy appears to have been given to it from its use. The 8th definition is left out; because it does not belong to the Elements ; nor can it be explained, so as to be understood by beginners, as is observed by Dr Simson. The 13th definition is also omitted, because it is useless in a translation, its only design being to explain a Greek word. And the 19th, which is the definition of a segment, 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, be cause it is very frequently used by Geometers. These are all the alterations that have been made in the defi. nitions of this book ; but many more might have been made with propriety. The first nine definițions might have beer given in the form of an introduction, for they are none of them geometrical, except the seventh, as amended by Dr Simson. The terms by which a line and a superficies 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 considering their relation to a solid, and to one another, as Dr Simson has done. M m 2 be a part BOOK I. “It is neceffary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand a right the definitions of a point, line, and superficies; for all these arise from a solid, and exist in it: The boundary, or boundaries which contain a solid, are called superficies, or the boundary which is common to two solids, 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 solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other folid, it is called a superficies, and has no thickness : For if it has any, this thickness inuft either of the thickness of the solid AG, or the folid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the folid BM; becaute, if this folid be removed from the solid AG, the superficies BCCF, the boundary of the folid AG remains still the same as it was. Nor can it be a part of the thickness of the solid AG; because, if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain ; therefore the superficies BCGF has no thickness, but only length and breadth. The boundary of a superficies is called a line; or a line is the common boun I dary of two fuperficies that are conti 71 N guous; or it is that which divides one fuperficies into two contiguous parts : Thus, if BC be one of the boundaries which contain the superficies ABCD, A B or which is the common boundary of this fuperficies, and of the superfícies KBCL, which is conti guous to it, this boundary BC is called a line, and has no breadth : For, if it has any, this must be part eicher of the breadth of the superficies ABCD, or of the superficies KBCL, or part of each of them. . It is not part of the breadth of the fuperficies KBCL ; foc, if this fuperficies be removed from the Superficies ABCD, the line BC, which is the boundary of the superficies ABCD, remains the same as it was : Nor can the breadth that BC is supposed to have be a part of the breadth of the superficies ABCD, because, if this be removed from the superficies KBCL, the line BC, which is the boundary of the fuperficies KBCL, does nevertheless remain ; therefore the line BC has no breadth: and because the line BC is in a superficies, and that a superficies has no thickness, as was thewn; therefore a line has neišlier breadth nor thickness, but only length. 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. No definition of a straight line has been given that is unexcep- Plato's definition, that the extremity of a straight line casts a One of them is, that a straight line is that of which and to two. |