Here 30° 00' is the mean Arc, whofe Sine is 500000 00000, and the Sine of 29° 59, the given Extream is ,49974806226, and the Length of the Arc z, viz. one Minute is ,000 29 0888208, which fquar'd and multiplied by the Sine of the mean Arc, 50000, &c. according to the Direction of the Theorem, the Product will be the fecond Difference, equal to ,000000042307, which fubtracted from double the Sine of the mean Arc, equal to 1, the Remainder will be,'999999957693, from which subtract the Sine of the given Extream (which in this Cafe is the leaft) and there will remain,5002518943 for the Sine of 30' 01', the greater Extream. This Method of making the Sines, however it may appear at firft Sight, is fo far from being tedious or troublesome, that I look upon it to be the moft eligible of any other whatsoever; for the Square of z being once determined, and the feveral Multiples of it by the nine Digits made, and fet down in a Table orderly, all the Sines may be made by Addition and Subtraction only; as indeed our Author hints they may by the Me thod demonftrated in the 10th Propofition of the Elements of Trigonometry; but this is evidently preferable to that, tho' a good Method too; and by which, all the Sines of the Quadrant, I prefume, were wont to be made, at least as far as 30, or 60 Degrees; for after the Sines as far as 60 Degrees are obtain'd, all the others may be had by Addition only; and notwithstanding there are other excellent Theorems, which contribute very much towards finishing and confirming the Truth of the whole Canon; yet this deduc'd from our Author's Series, I deem the most elegant and fit for Practice; because the Difference of the Differences of the Sines, being what is always required to be found, there will be feven Cyphers at least before the fignificant Figures of the faid Difference; which is the Product made by the Square of z, into the Sine of the mean Arc: So that to have the Sine true to ten Places, there will not be occafion to find above four or five Figures in the Product, which according to the common Method of contracted Multiplication, may be obtain'd with very few Figures. Thus, for inftance, the Sine of 30 02 may be had to ten Places by a wonderful eafy Operation, the Sines of 30° 01' and 30° oo' being both given. EXAMPLE. The Sine of 30° or is ,50025189543 40020 2001 300 5 42326 Whence the Product is ,000000042326 true to eleven Places at least. Wherefore, if according to the Rule, from double the Sine of the middle Arc=1,00050379086 we fubtract the faid Product, And from the Remainder the Sine of 30° oo the given Extream be fubftracted ,00000004232 1,00050374854 50000000000 50050374854 There will remain,50050374854 for the Sine of 30° 02' the other Extream; than which, nothing of this Nature can be defired more easy. SCHOLIUM. Because the Difference of the Differences of the Sines, or fecond Difference, has always 7 Cyphers before the fignificant Figures; it follows, that the whole Canon where the Sines confift but of 6 Places, which is as far as our Tables for common Practice need extend, may be perform'd chiefly by Addition and Subtraction only, without forming Multiples of the Square of z by the nine Digits; tho' perhaps it may be neceffary to use the Method of contracted Multiplication every 5th Minute to confirm the Truth, left in continual doubling and subtracting an Error fhould arife in the right-hand Figure however, as it may be fafely used for 5 Minutes together, and fometimes more, it will render the whole very easy. Note, the Square of z in this Cafe, viz. the Arch of 5 Minutes is 00000211. Thus Thus having investigated the Newtonian, and our Author's Series, and exemplified the latter, by making the Sines of 30° 01' and 30° oz', and withal fhewn how from the Sine of an Arc given, to find the Length of that Arc, and confequently the Circumference of the whole Circle/I fhall beg leave before I treat of the Conftruction of Logarithms, to fhew how from the known Ratio of the Diameter to the Circumference, or any other Ratio whatsoever, that a Set of integral Numbers may be found, whofe Ratios fhall be the neareft poffible to the Ratio given; for which I hope to be excus'd, and the rather, because I believe this Method of determining them, was never before publish'd. RULE. Divide the Confequent by the Antecedent, and the Divifor by the Remainder, and the last Divisor by the laft Remainder, and fo on till nothing remains. Then for the Terms of the first Ratio, Unity will always be the Antecedent, and the firft Quotient the firft Confequent. For the TERMS of the fecond RATIO. Multiply the laft Antecedent by the second Quotient, and to the Product add) Nothing and fo will the Re fult, be the second Unity SAntecedent For all the following RATIOS. Multiply the laft { Antecedent by the next Quotient, Confequent and to the Product add the laft Antecedent? 2 Confequent S and fo will the Sum be the prefent Antecedent Confequent but one; EXAM EXAMPLE. Let it be required to find a Rank of Ratios, whofe Terms are integral; and the neareft poffible to the following Ratio, viz. of 10000 to 31416, which expreffes nearly, the Proportion of the Diameter of the Circle, to its Circumference. But because the Terms of the Ratio are not prime to each other, they muft therefore be reduc'd to their leaft Terms. Whence 10000 1250 and then 3927 divided by 31413917 1250 and 1250 by the Remainder, &c. will be as follows. 44416 So the first Antecedent is 1, and the firft Confequent 3. Anteced.12 Confeq. 3}x7={{7+ 7 the 2d Anteed. 21221+1=22 the zd Confq. Which 7 and 22 is Archimedes's Proportion. I Anteced. 7X 16 = { 112, 112+ Confeq. 22 113 the 3d Anteca. 2352 352+3=355 the 3d Confeq. Which Terms 113 and 355 is Metius's Proportion. S Antecd. 113 S1243 S124371250 23905423905+22=3927 Producing the fame Antecedent and Confequent as at firft; which as it is ever the Property of the Rule fo to do, proves at the fame time, that no Error has been committed thro' the whole Operation. Whence as 1250:3927 1:3 7:22 For the I >Terms 113:355 of the Ratio. But it muft be obferv'd that 1 to 3 does not express the Ratio fo near as 7 to 22, nor 7 to 22 fo near as 113 to to 355, that is, the larger the Terms of the Ratio are, the nearer they approach the Ratio given. Mr. Molyneux, in his Treatife of Dioptricks informs us, that when Sir Ifaac Newton fet about by Experiments, to determine the Ratio of the Angle of Încidence, to the refracted Angle, by the Means of their respective Sines; found it to be from Air to Glafs, as 300 to 193, or in the leaft round Numbers, as 14 to 9. Now if it be as 300 to 193, it will readily appear by the Rule, whether they are such integral Numbers, whose Ratio is the neareft poffible to the given Ratio. 193) 300 (1 86) 107. (I 88 21) 8 (4 For dividing the greater Number by the lefs, and the lefs by the Remainder, &c. the Operation will fhew that the Numbers 193 and 300 are Prime to each other; and that the firft Antecedent is 1, as alfo the firft Conходили деловит Whence I And { I the fecond Antecedent 2 the second Confequent. Again {}={And +2 the third Antecedent 2 3 the third Confequent. Again {3}x4={12 I And 819 the 4th Antecedent 12214 the 4th Confequent. Hence, the fourth Antecedent and Confequent, make and 143; Sir Ifaac Newton himself, determines the Ratio out of Air into Glass to be as 17 to 11; but then he speaks of the Red Light, For that, great Philofopher in his |