Equity from the Operations, as ought to preside in all Contracts of this Nature. Whence it follows, that all other Methods, whose Resolution differs from this (especially if the Difference be much), may justly be deemed erroneous, and consequently prejudicial to one of the Parties concerned. Wherefore, to prevent Impofitions thro' Ignorance, great Care should be taken; which Precaution, however unnecessary it may appear, 'tis presumed, it will be regarded, inasmuch as no one is willing to pay more Years Purchase than he has Chances for living; as, on the contrary, the Seller to receive less than his Due; which may possibly be by following the common Methods (where, for the most part, Regard is had neither to Age nor Interest) founded upon Caprice, Humour, or, if you please, Custom, the Contract being made, as they can agree, right or wrong; which Method of Procedure ought to be exploded, since so liable to Error, and the Consequences drawn therefrom so often wide of the Truth. The other Instance which I shall give of the great Ufe of Logarithms is in the Case of Seffa, as related by Dr. Wallis in his Opus Arithmeticum from Alfephad (an Arabian Writer), in his Commentaries upon Tograius's Verfes; namely, that one Seffa, an Indian, having first found out the Game at Cheffe, and shewed it to his Prince Shebram, the King, who was highly pleased with it, bid him afk what he would for the Reward of his Invention; whereupon he asked, That for the first little Square of the Cheffe Board, he might have one Grain of Wheat given, for the second two, and fo on, doubling continually, according to the Number of the Squares in the Cheffe Board, which was 64. And when the King, who intended to give a very noble Reward, was much displeased, that he had asked so triAing an one, Seffa declared, that he would be contented with this small one. So the Reward he had fixed upon was ordered to be given him. But the King was quickly aftonished, when hefound, that this would rise to so vast a Quantity, that the whole Earth itself could not furnish out fo much Wheat. But how great the Number of these Grains is, may be found by doubling one continually 63 Times, so that we may get the Number that comes in the last Place, and then one Time more to have the Sum of all; for the Double of the last Term less 1 by one is the Sum of all. Now this will be most expeditioufly done by Logarithms, and accurately enough too for this Purpose: For if to the Logarithm of I, which is o, we add the Logarithm of 2 (which is 0,3010330) multiplied by 64, that is, 19,2659200, the abfolute Number agreeing to this will be greater than 18446, 00000, 00000, 00000, and less than 18447, : 00000, 00000, 00000. As I have had the revising of these Sheets, so it may be expected that I should give my Opinion concerning Mr. Cunn and our Author, in regard to Spherical Trigonometry; wherein the former accuses the latter, and several other eminent Authors, of having committed many Faults, and, in some Cafes, of being mistaken, especially in the Solution of the 12th Cafe of Oblique Spherics; in which Mr. Cunn has intirely mistaken the Author's Meaning, as plainly appears by his Remark, where he constitutes a Triangle whose Sides are equal to the given Angles; whereas the Author means, that each Angle should first be changed into its Supplement, and then with the said Supplements another Triangle constituted, whole Angles, by the very Text of the 14th Propofition of his own Spherical Trigonometry, will be the Supplements of the Sides fought in the given Triangle; to which Proposition I refer the Reader. That this is the Sense of the Author, is very evident, if impartially attended to, and which I think could poffibly have no other Meaning; and accordingly aver what is here advanced to be universally true; but, because I would not be misunderstood, shall illustrate the Truth thereof by a numerical Operation; which, to those who care not to trouble themselves with the Demonstration, may be fufficient; and, to others, some Satisfaction. EXAMPLE. Suppose, in the Oblique-angled Spherical Triangle ADE, there are given the Angles A, D, E, as per Figure, and the Side DE required. Note, Write down the Supplements of the two Angles next the Side required first; and then the Operation may stand thus: The Sum, minus CE = 120 the Supplement of the Angle. { D= 20 Which last Figures 9,944179 give the Side of 61° 34'; and the Double thereof, viz. 123° 08′, fubtracted from 180 Degrees, leaves for the Supplement 56°52', which is the Side of DE required. The Rule which Mr. Cunn substitutes in the Room of our Author's, is also universal (but not new); and, consequently, when he says, Change one of the Angles adjacent to the Side fought into its Supplement, it is very juft; though, by the way, I affirm, it is equally true, if the Angle oppofite to the Side fought were changed into its Supplement (which perhaps is what has not yet been taken Notice of); only then, instead of having the Side fought directly, we should have its Complement to 180 Degrees, as in the preceding Example; but there is a Necessity of changing either one or all the Angles into their Supplements, though it is beft to change only one, which let be either of those next the Side fought, no matter which; and the Side will be had directly without any Subduction, as will appear by the subsequent Operation. 45 EXAMPLE. Let the Angle E be changed into its Supplement, and the Side DE fought; which Supplement, and the other Angle adjacent to the Side sought, being written down first, the Operation may be as follows: Cc 4 other Sup. of the Angle E=50 Sum minus The D=30 Sum 120 2 Sum 60 Sup. Angle E = 10 Sine Co. Ar.-0,301030 Sine Co. Ar.-0,115746 Angle D =30 Sum 19,355416 Sum 9,677708 Which half Sum 9,677708 gives the Sine of 28° 26', and the Double thereof 56° 52' is the Side DE fought, the fame as before, when all the Angles were changed into their Supplements. Whence it is abundantly manifest, that those two Methods of Operation, notwithstanding their Manner is so different, agree precisely in Practice; and, confequently, we may conclude our Author's Rule to be right. Wherefore I wonder Mr. Cunn did not attend better to the Words of our Author's Rule, before he ventured to attack the Characters of so many famous Trigonometrical Writers. But to remove the Imputation of the Charge against those Authors who have deserved so well of the Mathematics, and to justify them to the World (for Justice ought to have Place), it is, that I have ventured to give my Opinion, and point out where Mr. Cunn was mistaken: The Reason of which is not easily affigned, fince, to give him his Due, it could not be for want of Knowledge, tho', in this Cafe, I can't think it intirely owing to Inadvertence, inasmuch as it was a premeditated Thing; and I am loth to impute it to any contentious Inclinations of his, in difputing the Veracity of our Author's Rule, because it did not appear with all that Plainness requifite to prevent carping by the Litigious: Wherefore, as I am in Suspense how to determine, I shall leave the Decifion thereof to better Judgments. Indeed, Mr. Heynes's Rule, which directs with the three Angles given to project a Triangle, as if they were Sides, is deficient, were it only on that very Account: For with the given Angles, in the preceding Example, Example, it will be impossible to construct a Triangle, because 'tis requifite, that two Sides together, however taken, be greater than the third; whereas, in this Cafe, they will be less: But the Rule is not only deficient in that Respect, but really wrong: For tho' what Mr. Heynes afferts is just, viz. that the greatest Side in the supplemental Triangle is the Supplement of the greatest Anglein the other Triangle; yet, notwithstanding that, the Consequence drawn therefrom is false, and so the Solution only imaginary: For, with Submiffion, neither the Sides, nor their Supplements, in Mr. Heynes's fupplemental Triangle, are the Measures of the Sides fought. 'Tis true, when one of the Angles is a Right one, and the others both acute, then the said fupplemental Triangle is that wanted to be constructed, as containing all the given Angles; and, confequently, the Sides appertaining thereto are the very Sides required: But then this is only one instance out of the infinite Number of other Triangles that may be conftructed, and which is not solved directly by the Triangle first projected neither; for the greatest Angle thereof must be changed into its Supplement, when the Side oppofite to the Right Angle is requir'd; and if the Right Angle still remains, and either one or both of the other given Angles are obtuse, the Solution is render'd more perplex'd: Wherefore there can be no general Solution given to any Triangle, by constituting a Triangle whose Sides are equal to the given Angles, except to that particular one which Mr. Gunn takes Notice of in his Remark, where each given Angle is the Measure of its oppofite Side fought, and which therefore needs no Operation. This I thought myself obliged to observe, in Justice to Mr. Cunn, who, we fee, is not intirely to blame; as having just Reason to object against the Veracity of Mr. Heynes's Rule, tho' not againft the Rules of the other Authors by him nominated. And here I can't but take Notice of some Gentlemen, who are so very fond of finding Fault, that, rather than you shall not be in the Wrong, they will wrest your own Meaning from you, and will not fuffer an Error, tho' ever so minute, to pass, without proclaiming it to the Public, under Pretence of preventing their being impos'd upon; whereas, if the Truth were known, I fear it would appear to be the Vanity of their Hearts, an 1 |