Which laft 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 fubftitutes in the Room of our Author's, is alfo univerfal (but not new); and, confequently, when he fays, 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 Neceffity of changing either one or all the Angles into their Supplements, though it is beft to change only one, which let be either of thofe next the Side fought, no matter which; and the Side will be had directly without any Subduction, as will appear by the fubfequent Operation.

EXAMPLE.

Let the Angle E be changed into its Supplement, and the Side D E fought; which Supplement, and the

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other

other Angle adjacent to the Side fought, being written down first, the Operation may be as follows:

Sup. of the Angle E-50

The SD=30

Angle A=40

Sum 120
2|-

Sum 60

Sine Co. Ar.-0,115746

Sine Co. Ar.-0,301030

Sine 10°

9,239670

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 D E fought, the fame as before, when all the Angles were changed into their Supplements.

Whence it is abundantly manifeft, that those two Methods of Operation, notwithstanding their Manner is fo different, agree precifely 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 fo many famous Trigonometrical Writers. But to remove the Imputation of the Charge against thofe Authors who have deferved fo well of the Mathematics, and to juftify them to the World (for Juftice 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 eafily 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 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 Sufpenfe how to determine, 1 fhall leave the Decifion thereof to better Judgments.

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Indeed, Mr. Heynes's Rule, which directs with the three Angles given to project a Triangle, as if they were Sides, is deficient, weve it only on that very Account: For with the given Angles, in the preceding Example,

Example, it will be impoffible to construct a Triangles becaufe 'tis requifite, that two Sides together, however taken, be greater than the third; whereas, in this Cafe, they will be lefs: But the Rule is not only deficient in that Respect, but really wrong: For tho' what Mr. Heynes afferts is juft, viz. that the greateft Side in the fupplemental Triangle is the Supplement of the greatest Angle in the other Triangle; yet, notwithstanding that, the Confequence drawn therefrom is falfe, and fo 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 faid fupplemental Triangle is that wanted to be conftructed, as containing all the given Angles; and, confequently, the Sides appertaining thereto are the very Sides required: But then this is only one Inftance out of the infinite Number of other Triangles that may be conftructed, and which is not folved directly by the Triangle firft 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 ftill remains, and either one or both of the other given Angles are obtufe, the Solution is render'd more perplex'd: Wherefore there can be no general Solution given to any Triangle, by conftituting a Triangle whofe 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 oppofite Side fought, and which therefore needs no Operation,

;

This I thought myfelf obliged to obferve, in Juftice to Mr. Cunn, who, we fee, is not intirely to blame as having juft Reafon to object against the Veracity 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 fome Gentlemen, who are so very fond of finding Fault, that, rather than you fhall not be in the Wrong, they will wreft your own Meaning from you, and will not fuffer an Error, tho' ever fo 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,

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an Over-fondnefs of being thought wifer and more knowing than the reft of Mankind; nay, I think, it appears plainly fo, by their oppofing the Works of Men greater than themselves: But if, instead of comparing how far their finite Knowledge extend, or exceeded another Perfon's, they confider'd how much there was they knew nothing of; as it would conduce to make them humble, fo, I am of Opinion, it would contribute verymuch toward theirleaving off that Manner of Writing. Befides, as I take it, the Bufinefs of Writing is not fo much to difcover who has committed the moft Faults, as to avoid them, and make greater Improvements.

But, what is the moft to be wonder'd at, those who are so very ready in finding Fault, not without great Sufpicion, receive the best Part of their Knowledge from the Works of thofe very Authors against whom they exclaim. The Reason that induces me to think fo is this: Whilst they are ftudying an Author, in order to understand him, then it is, perhaps, they discover fomething which he was pleased to omit, or thought fit to conceal, for which 'tis more than probable they take Care not to omit paying a profound Refpect to their vainly-imagined fuperior Geniuses: And if, by Accident, an Error fhould creep in (which is very poffible, none being infallible), then, to be fure, he must be egregiously mistaken, and not understand what he was about: But, I fay, this Difquifition into the Demerits of an Author would never have been made, had they understood the Subject beforehand; for, if otherwise, they must be of a fad Cynical Temper, as well as have little elfe to do, to make it their Business to difcover Faults, and at the fame Time acknowledge not one fingle Beauty; a very ungrateful Return for the Advantage they receive in the Perufal.

Nor do they do the Public that Service they pretend to: For those that are capable, and will be at the Trouble, of reading a Treatife upon a Subject without a Mafter, are as well able as themselves to rectify what is amifs; and as for those who will not be at that Trouble, there is no Danger of their being led aftray; fince it is the fame Thing to them, whether there be any Mistakes, or not.

However, if, after all, there fhould be a Neceffity for an Admonition, why can't it be done with Candour and Humanity?

Humanity? And then, without doubt, an Author, out of Regard to Truth, which of all Things ought to be preferred, would be thankful: And to reprove otherwise, is to be ungenerous; because, whenever thofe Miftakes happen, as they are for the most part owing more to Inadvertency, than Want of Knowledge; fo they should therefore be attributed to the Frailty of human Nature (to which we are all more or less subject), nothing being more common amongst all Profeffions, than the writing of one Thing for another.

If any think, by my interfering between our Author and Mr. Cunn, that I have run into the fame Error, of which I accufe others in general of being guilty, let them please to confider that I have only writ in the Vindication of Gentlemen, who were firft wrongfully accus'd, and in one Particular juftify'd Mr. Cunn: For fuch an Occafion as this offering, I thought the Difference between them lay upon me to decide, left I should be taxed with Partiality for not doing Juftice, or with Ignorance in not determining an Affair which held fome in Sufpenfe to know who was in the right or wrong; for there could be no Poffibility of making a Merit in adjusting a Thing of fo eafy a Nature; tho', perhaps, to conceive thoroughly the Reafon of all the different Methods of Solution, may not be fo eafy neither.

But, to proceed: As for the Omiffions our Author has made in not determining accurately when fome of the Cafes are ambiguous, and when not, I fhall not quarrel with those who think him to blame; but, if I may be allowed to give my Opinion, I think they are determin'd for the most part, as well, or, at leaft, with more Eafe, from the Construction of the Triangles, because it fixes an Idea of what one is about, by exhibiting a kind of an ocular Demonftration; and, confequently, prevents the laying of that Strefs upon the Memory, as all thofe are obliged to who depend intirely upon Mr. Cunn's Rules, which to Beginners is not very agreeable: Hence, who knows but that what our Author wrote relating to the ambiguous Cafes, he thought fufficient? That is, that the Reader would not ftop, for want of farther Explications, but with more Ease supply himself with what was wanting when he came to the Practice thereof, I mean the Conftruction of Triangles (for, after all, without the Knowledge of that,

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