angle; and in extreme cases, that is, when the angle sought is very acute or very obtuse, this distinction is very material to be considered. The reason is, that the sines of angles, which are nearly = = 90°, or the cosines of angles, which are nearly = 0, vary very little for a considerable variation in the corresponding angles, as may be seen from looking into the tables of sines and cosines. The consequence of this is, that when the sine or cosine of such an angle is given (that is, a sine or cosine nearly equal to the radius,) the angle itself cannot be very accurately found. If, for instance, the natural sine .9998500 is given, it will be immediately perceived from the tables, that the arc corresponding is between 89°, and 89° l'; but it cannot be found true to seconds, because the sines of 89° and of 890 1', differ only by 50 (in the two last places,) whereas the arcs themselves differ by 60 seconds. Two arcs, therefore, that differ by 1", or even by more than 1", have the same sine in the tables, if they fall in the last degree of the quadrant. The fourth solution, which finds the angle from its tangent, is not liable to this objection; nevertheless, when an arc approaches very near to 90°, the variations of the tangents become excessive, and are too irregular to allow the proportional parts to be found with exactness, so that when the angle sought is extremely obtuse, and its half of consequence very near to 90, the third solution is the best. It may always be known, whether the angle sought is greater or less than a right angle by the square of the side opposite to it being greater of less than the squares of the other two sides. SECTION III. CONSTRUCTION OF TRIGONOMETRICAL TABLES. In all the calculations performed by the preceding rules, tables of sines and tangents are necessarily employed, the construction of which remains to be explained. The tables usually contain the sines, &c. to every minute of the quadrant from 1' to 90°, and the first thing required to be done, is to compute the sine of 1', or of the least arc in the tables. 1. If ADB be a circle, of which the centre is C, DB, any arc of that circle, and the arc DBE double of DB; and if the chords DE, DB be drawn, also the perpendiculars to them from C, viz. CF, CG, it has been demonstrated (8. 1. Sup.), that CG is a mean proportional between AH, half the radius, and AF, the line made up of the radius and the perpendicular CF. Now CF is the cosine of the arc BD, and CG the cosine of the half of BD; whence the cosine of the half of any arc BD, of a circle of which the radius = 1, is a mean proportional between and 1+cos. BD. Or, for the greater generality, supposing A = any arc, cos. A is a mean proportional between and 1+cos. A, and therefore (cos. A)2=(1+cos. A) or cos. } A=√1⁄2 (1+cos. A). 2. From this theorem, (which is the same that is demonstrated (8. 1. Sup.), only that it is here expressed trigonometrically,) it is evident, that if the cosine of any arc be given, the cosine of half that arc may be found. Let BD, therefore, be equal to 60°, so that the chord BD=radius, then the cosine or perpendicular CF was shewn (9. 1. Sup.) to be, and therefore cos. BD, or cos. 30°=√(1+})=√?= In the same man = √3 ner, cos. 15°=√(1+cos. 30°), and cos. 70, 30'=√(1+cos.15°), &c. In this way the cosine of 30, 45', of 10, 52′, 30′′, and so on, will be computed, till after twelve bisections of the arc of 60°, the cosine of 52′′. 44′′. 93""". 45. is found. But from the cosine of an arc its sine may be found, for if from the square of the radius, that is, from 1, the square of the cosine be taken away, the remainder is the square of the sine, and its square root is the sine itself. Thus the sine of 52". 44". 03""". 45. is found. 3. But it is manifest, that the sines of very small arcs are to one another nearly as the arcs themselves. For it has been shewn that the number of the sides of an equilateral polygon inscribed in a circle may be so great, that the perimeter of the polygon and the circumference of the circle may differ by a line less than any given line, or, which is the same, may be nearly to one another in the ratio of equality. Therefore their like parts will also be nearly in the ratio of equality, so that the side of the polygon will be to the arc which it subtends nearly in the ratio of equality; and therefore, half the side of the polygon to half the arc subtended by it, that is to say, the sine of any very small arc will be to the arc itself, nearly in the ratio of equality. Therefore, if two arcs are both very small, the first will be to the second as the sine of the first to the sine of the second. Hence, from the sine of 52′′. 54"". 03""". 45. being found, the sine of 1′ becomes known, for, as 52". 44". 03""". 45. to 1, so is the sine of the former arc to the sine of the latter. Thus the sine of 1' is found = 0.0002908882. 4. The sine l' being thus found, the sines of 2', of 3', or of any number of minutes, may be found by the following proposition. THEOREM. Let AB, AC, AD be three such arcs, that BC the difference of the first and second is equal to CD the difference of the second and third; the radius is to the cosine of the common difference BC as the sine of AC, the middle arc, to half the sum of the sines of AB and AD, the extreme arcs. D M Draw CE to the centre: let BF, CG, and DH perpendicular to AE, be the sines of the arcs AB, AC, AD. Join BD, and let it meet CE in I; draw IK perpendicular to AE, also BL and IM perpendicular to DH. Then, because the arc BD is bisected in C, EC is at right angles to BD, and bisects it in I; also BI is the sine, and EI the cosine of BC or CD. And, since BD is bisected in I, and IM is parallel to BL (2. 6.), LD is also bisected in M. Now BF is equal to HL, therefore BF +DH=DH+HL=DL+2LH = 2LM+ 2LH=2MH or 2KI; and therefore IK is half the sum of BF and DH. But because the triangles CGE, IKE are equiangular, CE: EI:: CG: IK, and it has been shewn that EI=cos. BC, and IK= (BF+DH); therefore R: cos. BC:: sin. AC: (sin. AB+sin. AD). COR. Hence, if the point B coincide with A, B AF GK H गु R: cos. BC sin. BC: sin. BD, that is, the radius is to the cosine of any arc as the sine of the arc is to half the sine of twice the arc; or if any arc-A, sin. 2A=sin. A× cos. A, or sin. 2A=2 sin. Ax cos. A. Therefore also, sin. 2′-2 sin. l' x cos. 1': so that from the sine and cosine of one minute the sine of 2' is found. Again, 1', 2', 3', being three such arcs that the difference between the first and second is the same as between the second and third, R: cos. 1' : : sin. 2(sin. l'+sin. 3'), or sin. 1'+sin. 3'2 cos. 1'+sin. 2', and taking sin. 1' from both, sin. 3'=2 cos. 1'x sin. 2'-sin. 1. In like manner, sin. 4'=2′ cos. 1'x sin. 3′-sin. 2, sin. 5'2' cos. 1'x sin. 4'-sin. 3, sin. 6'2' cos. 1'x sin. 5′-sin. 4, &c. Thus a table containing the sines for every minute of the quadrant may be computed; and as the multiplier, cos. 1' remains always the same, the calculation is easy. For computing the sines of arcs that differ by more than 1', the method is the same. Let A, A+B, A+2B be three such arcs, then, by this theorem, R cos. B:: sin. (A+B): (sin. A+sin. (A+2B)); and therefore making the radius 1, sin. A+sin. (A+2B)=2 cos. Bx sin. (A+B), or sin. (A+2B)=2 cos. Bx sin. (A+B)-sin. A. By means of these theorems, a table of the sines, and consequently also of the cosines, of arcs of any number of degrees and minutes, from 0 to 90, sin. A the table of tangents cos. A' may be constructed. Then, because tan. A= is computed by dividing the sine of any arc by the cosine of the same arc. When the tangents have been found in this manner as far as 45°, the tangents for the other half of the quadrant may be found more easily by another rule. For the tangent of an arc above 45° being the co-tangent of an arc as much under 450; and the radius being a mean proportional between the tangent and co-tangent of any arc (1. Cor. def. 9), it follows, if the difference between any arc and 45° be called D, that tan. (45°-D): 1 :: 1: tan. (45°+D), so that tan. (45°+D)=; 1 tan. (450-D) Lastly, the secants are calculated from (Cor. 2. def. 9.) where it is shewn that the radius is a mean proportional between the cosine and the secant of any arc, so that if A be any arc, sec. A= 1 cos. A The versed sines are found by subtracting the cosines from the radius. 5. The preceding Theorem is one of four, which, when arithmetically expressed, are frequently used in the application of trigonometry to the solution of problems. 1mo, If in the last Theorem, the arc AC=A, the arc BC=B, and the radius EC I, then AD=A+B, and AB-A-B; and by what has just been demonstrated, 1 cos. B sin. A: sin. (A+B)+ sin. (A-B), : sin. Ax cos. B and therefore, sin. (A+B)+ (A-B). 2do, Because BF, IK, DH are parallel, the straight lines BD and FH are cut proportionally, and therefore FH, the difference of the straight lines FE and HE, is bisected in K; and therefore, as was shewn in the last Theorem, KE is half the sum of FE and HE, that is, of the cosines of the arcs AB and AD. But because of the similar triangles EGC, EKI, EC : EI::GE: EK; now, GE is the cosine of AC, therefore, R: cos. BC cos. AC: cos. AD+ cos. AB, or 1: cos. B: cos. A: cos. (A+B)+ cos. (A—B); and therefore, cos. Ax cos. B= cos. (A+B)+cos. (A-B); 3tio, Again, the triangles IDM, CEG are equiangular, for the angles KIM, EID are equal, being each of them right angles, and therefore, taking away the angle EIM, the angle DIM is equal to the angle EIK, that is, to the angle ECG; and the angles DMI, CGE are also equal, being both right angles, and therefore the triangles IDM, CGE have the sides about their equal angles proportionals, and consequently, EC: CG :: DI : IM; now, IM is half the difference of the cosines FE and EH, therefore, R sin. AC sin. BC: cos. AB- cos. AD, or 1: sin. A :: sin. B: cos. (A-B)- cos. (A+B); and also, sin. Axsin. Bcns. (A-B)- cos. (A+B). 4to, Lastly, in the same triangles ECG, DIM, EC: EG :: ID: DM; now, DM is half the difference of the sines DH and BE, therefore, R: cos. AC sin. BC: sin. AD- sin. AB, or 1 cos. A sin. B: sin. (A+B)- sin. (A+B); and therefore, cos. Axsin. B= sin. (A+B)—sin. (A-B). 6. If therefore A and B be any two arcs whatsoever, the radius being cos. (A+B). sin. (AB). IV. cos. Ax sin. B= sin. (A+B) From these four Theorems are also deduced other four. sin. Ax cos. B+cos. Axsin. B=sin. (A+B). 7. Again, since by the first of the above theorems, sin. AX cos. B sin. (A+B)+ sin. (A-B), if A+B=S, and A-B=D, S-D 2 sin. S+D. But as S and D may be any arcs whatever, to preserve the former notation, they may be called A and B, which also express any arcs whatever: thus, sin. A+B X cos. A-B = sin. A+ sin. B, or In all these Theorems, the arc B is supposed less than A. 8. Theorems of the same kind with respect to the tangents of arcs may be deduced from the preceding. Because the tangent of any arc is equal to the sine of the arc divided by its cosine, |