EXAMPLE I. If the diameter of a circle be 7, what is the circumference? 31416 7219912 the circumference, nearly. 22 Therefore the diameter is to the circumference, nearly as 7 to 22, E X plaining it as follows. Since the chief advantage confifts in taking fmall arcs, whofe tangents fhall be numbers easy to manage, Mr. Machin very properly confidered, that fince the tangent of 45 is 1; and that the tangent of any are being given, the tangent of double that are can easily be had; if there be affumed fome fmall fimple number for the tangent of an arc, and then the tangent of the double arc be continually taken, until a tangent be found nearly equal to 1, the tangent of 452; by taking the tangent aniwering to the fmall difference between 45° and this multiple, there would be had two very fmall tangents, viz. the tangent first affumed, and the tangent of the difference between 45° and the multiple arc; and that, therefore, the lengths of the arcs, correfponding to these two tangents being calculated, and the arc belonging to the tangent firft affumed being as often doubled as the multiple directs, the refult increafed or diminished by the other arc, would be the arc of 45, according as the multiple are fhould be below or above it. Having thus thought of his method, by a few trials he was lucky enough to find a number, and perhaps the only one, proper for this purpofe, viz. knowing that the tangent of of 45° is nearly , he affumed as the tangent of an are: then, fince if t be the tangent of an arc, the tangent of the double arc will the radius being 1; the tangent of an arc double to be 1 2 t 25 119' I = which, being very nearly equal to 1, fhews that the arc which is equal to 4 times the firft, is very near 45°. Then, fince the tangent of the difference between 45° and an are whofe tangent is EXAMPLE II. What is the diameter of the circle whofe circum ference is 355? 35531416112.9997+, the diameter, 113 nearly. And therefore the diameter is to the circumference nearly as 113 to 355. Now, by calculating, from the general feries, the arcs whofe tangents are and, which may be very quickly done, by reafon of the fallnefs and fimplicity of the numbers, and taking the latter are from 4 times the forme, the remainder will be the arc of 45°. And this is Mr. Machin's famous quadrature of the circle. But it was by means of Dr. Halley's method that he found the circumference of a circle, whofe diameter is 1, to be 3.1415926535, $9793-3846,2643383279, 5028841971, 6939937510, 5820974944, 5923071864,06-8020899, 8628034825,3421170679 +, true to above 100 places of figures. Or, by fubftituting the above numbers in the general feries, 16 4 we get the feries( I 16 1.16 4 -- ) ( - 5 239 3 53 4 equal to the femicircumference whofe radius is 1, or the whole circumference whole diameter is 1. And this is the feries publifhed by Mr. JONES, and which he acknowledges he received from Mr. MACHIN. But because the arc whofe tangent is, is 2 times the arc whofe tangent is, minus the arc to tangents; (for tang. of dif. between the arcs whofe tangents are 30 and }); therefore 8 times are to tang. I' - 4 times arc to tang. to tang. = arc of 45°, or whofe tang, is 1. I - arc Which is much eafier than Machin's way. And various other ways may easily be difcovered from the fame principles. If instead of t, in the feries t(1 X substituted its value —, we shall have — × (1 – Τ T &c) for the length of the arc whofe co-tangent is 7, and radius r. Or, What is the circumference of the earth, fuppofing it to be perfectly round, and that its diameter is 7958 Anf. 25000 8528 miles. miles? 3c2 52 54 5c4 &c) 7c6 for the length of the arc whofe fine is s and cofine c. And, farther, by writing rr-ss for cc in this laft feries, we ob 52 tain s × (1+ 3.556 2.372 2.4.5,4 ' 2.4.6.7,5 2.4.6.8.9 78 the arc whofe fine is s. Again, by fubftituting /2rv-vv, the value of s, inftead of it, in the last feries, we get 322 + for the length of the arc whofe verfed fine is v.-Or, by writing d, the diameter, instead of 2r, the value of the arc will be ex མ preffed by dv× (1+ 2.31+ 3v2 +-3·573 2.3d 2.4.5d2 2.4.6.7d3 &c). Much after the fame manner the are may be expreffed by the fecant, co-fecants, or co-lines. And if for any of the letters, in the general feries, be fubftituted any of their particular values, as was done for the tangent, we may, by those means, obtain several different particular forms of infinite feries, whofe fums will be expreffed by means of the circumferences of any given circles: Thus, if, in the laft feries but one, s be taken, or the arc be fuppofed to be a quadrant, we fhall obtain the infinite feries 325 3.577 3 3-5 r+ + + + 2.༣༡༩༧ 2.4.54 2.4.6.76 2.3 2.4.5 2.4.6.7 r3 I -&c, orrx (1+: + &c), whofe fum will be equal to of the circumference of the circle whofe radius is r. And, if from this feries be taken the former feries, expreffed bys, there will refult EXAMPLE IV. If the circumference of the earth be 25000 miles, what is the diameter ? Anf. 7957 744 miles. . To find the Length of any Arc of a Circle RULE 1.* As 180 is to the number of degrees in the arc, So is 31416 times the radius, to its length. Or as 9 is to the number of degrees in the arc, EXAMPLE. Required the length of the arc ADB, whofe chord AB is 6, the radius being 9. By trigonometry, 9=AC: 3= D AP :: 1 = rad. of the tables : 3 3333333 the fine of the ACP, or of the arc AD, to the radius 1 ; and the degrees, in the table of fines, anfwering to this fine, are 19:4712206, the double of which is 38.9424412, the degrees in the whole arc ADB. Then, by the rule, 38.9424412 X 0174533 × 9 = 6.117063, the length of the arc required. RULE * DEMONSTRATION. For, when the radius is 1, half the circumference is 3.14159265 3.14159265 &c 180 degrees &c, and therefore 0174532925199 &c is the length of an arc of 1 degree; hence rx 01745&c = the length of 1 degree to the radius r, and, confequently, 01745&c xrx number of degrees in any arc, will be the length of that arc. 2. E. D. RULE II. *If d denote the diameter of the circle, and v the verfed fine of half the arc; the arc will be expreffed 302 3.503 + 2.3d 2.4.5d2 2.4.6.7d3 B+ 5.59 c&c. puttingq= and 2d√9+2A+ 3.39 2.3 4.5 6.7 A, B, C, &c, for the firft, fecond, third, &c, terms. EXAMPLE. Let there be taken here the fame example as before. Then PC✓CA - AP2 = √92-32 = √72= 62, and PD DCCP=9-6√251471862 = =v; also·51471862 ÷ 18·02859548=2=q• Hence A 2d√q = 6·087672 If r be the radius, and s the fine or half chord AP of the arc, then the length of the arc ADB will be 354 3.5.55 3.39 5·52c+ A+ B+ 6.7 &c,) 7.72 D &C, 8.9 ; and A, B, C, &c, are the preceding terms. *This rule is proved in page 122. Proved in page 122. E X This likewife is |