« ΠροηγούμενηΣυνέχεια »
deem'd erroneous, confequently prejudicial to one of the Parties concerned. Wherefore to prevent Impositions thro' Ignorance great Care should be taken ; which Precaution, however unnecessary it may appear, 'tis presum’d 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 poffibly be by following the common Methods, where for the most part regard is had neither to Age nor Interest, but founded upon Caprice, Humour, or if you please Custom, the Contract being made as they can agree right or wrong, which Method of Procédure 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 Use 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 Verses, namely that one Sessa an Indian have ing first found out the Game at Chelse, and thewed it to his Prince Shehram, the King, who was highly pleas'd with it, bid him ask what he would for the Res ward of his Invention; whereupon he ask'd, that for the first little Square of the Chesse Board, he might have one Grain of Wheat given, for the second two, and so on doubling continually, according to the Number of the Squares in the Chelė Board, which was 64. And when the King, who intended to give a very noble Reward, was much displeas'd that he had ask'd so triAing a one, Sefa declar'd, that he would be contented with this small one. So the Reward he had fix'd upon was order'd to be given him: But the King was quickly aftonish'd when he found that this would rise to so vait a Quantity, that the whole Earth itself could not furnish out so 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 by one is the Sum of all. Now this will be more expeditiously done by Logarithms, and accurately enough too for this purpose. Forlif to the Logarithm of: which is o, we add the Logarithm of 2, which is
0,3010300 multiplied by 64, that is 19,2659200 the absolute Number agreeing to this, will be greater than 18466, 00000, 00000, 00000, and less than 18447, 00000, 00000, 00000.
As I have had the revising of these Sheets, fo 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 Cases of being mistaken, efpecially in the Solution of the 12th Cafe of Oblique Sphericks, in which Mr. Cunn has entirely mistook 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 chang’d into its Supplement, and then with the said Supplements another Triangle conftituted, whose Angles by the very Text of the 14th Proposition 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 poflibly have no other Meaning ; and accordingly aver what is here advanc'd 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 theinselves with the Demonstration may be sufficient, and to others fome Satisfaction.
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 Supple- (E = 50° Sine Co. Ar. 0, 115746 ment of the D=150 Sine Co. Ar.,301030 Angle. -A=140 Sine 120°
9,937530 Sum=340 Sine 20 -9, 534052 21
19,888358 Sum, minus CE=120 the Supple
=12 ment of the
9, 944179 Angle
Which last Figures 9.944179, give the Sine of 61° 34' and the double thereof, viz. 123° 08' fubtracted from 180 Degrees, leaves for the Supplement 56° 52, which is the Side DE required.
The Rule which Mr. Gunn 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. Tho' by the way I affirm, it is equally true, if the Angle opposite 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 sought directly, we should have its Complement to 180 Degrees, as in the precedent Example; but there is a Necessity of changing either one, or all of the Angles into their Supplements, tho' it is best 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 Şubduction, as will appear by the subsequent Operation.
E X A M P L E.
Let the Angle E be changed into its Supplement, and the Side D E lought; which Supplement, and the oCc2
ther Angle adjacent to the Side fought, being writ down first; the Operation may be as follows. Sup. of the Angle E=50
Sine Co. Ar-0,115746 The SD=30
Sine Co. Ar-0,301030 Angle TA=40 Sine 10 9,239670 Sum 120
9,698970 21 Sum 60
Sum 19,355416 Sum 5 Sup. Angle E = 10
1 Sum 9,677708 mínus 2 Angle
D = 30
Which half Sum 9,677708 gives the Sine of 28° 56', and the double thereof 56° 52' is the Side D E sought, the same as before, when all the Angles were chang’d 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 consequently 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 ventur'd to attack the Characters of so many famous Trigonometrical Writers. But to remove the Imputation of his Charge against those Authors who have deferv’d so well of the Mathematicks, and to justify them to the World, for Justice ought to have place, it is, that I have ventur'd to give my Opinion, and point out where Mr. Cunn was mistaken: The Reason of which is not easily afsign'd, since to give him his due it could not be for want of Knowledge, though in this Cafe I can't think it entirely owing to Inadvertency, inasmuch as it was a premeditated Thing, and I'm loth to impute it to any contentious Inclinations of his, in disputing the Veracity of our Author's Rule, because it did not appear with all that Plainness, requisite to prevent carping by the Litigious ; wherefore as I am in Suspence how to determine, Ihall leave the Decifion thereof to better Judgments.
Indeed, Mr. Haynes'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 Accounts for with the given Angles in the preceding
Example, it will be impossible to construct a Triangle, because 'tis requisite 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 asserts is just, viz. that the greatest Side in the supplemental Triangle, is the Supplement of the greatest Angle in the other Triangle ; yet, notwithstanding thats the Consequence drawn therefrom is false, and so the Solution only imaginary: for with Submission, neither the Sides nor their Supplements in Mr. Heynes's Supplemental Triangle, are the Measures of the Sides sought: 'Tis true, when one of the Angles is a right one, and the others both acute, then the said Supplemental Triangle is that wanted to be constructed, as containing all the given Angles; and consequently 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 constructed ; 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 oppo. fite to the right Angle is required; and if the right Angle still remains, and either one or both of the other given Angles are obtuse, the Solution is rendered more perplext: Wherefore there can be no general Solution given to any Triangle, by conftituting a Triangle, whose Sides are equal to the given Angles; except to that particular one which Mr. Cunn takes notice of in his Remark, where each given Angle is the Measure of its opposite Side fought, and which therefore needs no Operation.
This I thought my felf obliged to observe, out of Justice to Mr. Cunn, who we see is not intirely to blame; as having just Reason to object against the Ve racity of Mr. Heynes's Rule, tho' not against 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 suffer an Error, tho' ever so minute, to pass, without proclaiming it to the Publick, 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,