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other Angle adjacent to the Side fought, being written
Sine Co. Ar.-0,301030
+ Sums Sup. Angle E = 10
Which half Sum 9,677708 gives the Sine of 28° 26', and the Double thereof 56° 52' is the Side D E fought, the same 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 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 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 assigned, fince, to give him his Due, it could not be for want of Knowledge, tho', in this case, 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 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 Sufpenfe how to determine, I shall leave the Decision 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, it will be impossible to construct a Triangle, becaufe 'tis requisite, that two Sides together, however taken, be greater than the third; whereas, in this Case, 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 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 Submission, 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
a one, and the others both acute, then the said fupplemental 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 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 whole Sides are equal to the given Angles, except to that particular one which Mr. Cunn takes Norice of in his Remark, where each given Angle is the Measure of its opposite Side sought, and which therefore needs nu Operation.
This I thought myself obliged to observe, in Justice to Mr. Cunn, who, we see, is not intirely to blame; as having just Reason 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 some Gentlemen, who are lo 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 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,
Over-fondness of being thought wiser and more knowing than the rest of Mankind; nay, I think, it appears plainly so, by their opposing the Works of Men greater than themselves : But if, instead of comparing how far their finite Knowledge extended, or exceeded another Person's, they consider'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 very much toward their leaving offthat Manner of Writing. Besides, as I take it, the Business of Writing is not so much to discover who has committed the most Faults, as to avoid them, and make greater Improvements.
But, wbat is the most to be wonder'd at, those who are to very ready in finding Fault, not without great Suspicion, receive the best part of their knowledge from the Works of those very Authors against whom they'exclaim. The Reason that induces me to think so is this: Whilft they are studying 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 it is more than probable they take Care not to omit paying a profound Respect to their vainly-imagined fuperior Geniuses : And if, by Accident, an Error should creep in (which is very possible, none being infallible), then, to be fure, he must be egregiously mistaken, and not understand what he was about: But, I say, this Disquisition 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 sad Cynical Temper, as well as have little else to do, to make it their Business to dircover Faults, and at the same Time acknowledge not one single Beauty; a very ungrateful Return for the Advantage they receive in the Perusal.
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 Treatise upon a Subject without a Mafter, as are 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 astray; since it is the same Thing to them, whether there be any Mistakes, or not.
However, if, after all, there should be a Necessity for an Admonition, why can't it be done with Candour and
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 those Mir takes happen, as they are for the most part owing more to Inadvertency, than Want of Knowledge ; so 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 Profeflions, 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 same Error, of which I accuse others in general of being guilty, let them please to consider that I have only writ in the Vindication of Gentlemen, who were first wrongfully accus'd, and in one Particular justify'd Mr. Cunn: For such an Occasion as this offering, I thought the Difference between them lay upon me to decide, lest I should be taxed with Partiality for '
not doing Justice, or with Ignorance in not determining an Affair which held some in Suspense to know who was in the right or wrong; for there could be no Possibility of making a Merit in adjusting a Thing of so ealy a Nature ; tho', perhaps, to conceive thoroughly the Reason ofallthedifferent Methods of Solution, may not be so easy neither.
But, to proceed : As for the Omissions our Author has made in not determining accurately when some of the Cases are ambiguous, and when not, I shall 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 least, with more Ease, from the Construction of the Triangles, because it fixes an Idea of what one is about, by exhibiting a kind of an ocular Demonstration; and, consequently, prevents the laying of that Stress upon the Memory, as all those 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 stop, 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 Construction of Triangles (for, after all, without the Knowledge of that,
a person will have but a mean Notion of this useful Branch of the Mathematics); and, if so, he ought in some measure to be excused, especially if to this we join the following Confideration, viz. that few or none ever learn Spherical Trigonometry, purely for the Sake of calculating Sides and Angles, to determine their Ambiguities; besides what is ambiguous in Trigonometry, is very often not so in Geography and Astronomy, &c. for which the other is chieây learnt.
For Instance : If we know the Latitude of London, and the Distance and Difference of Longitude between the faid Place and Rome, notwithstanding there are two Sides, and the Angle opposite to one of them, given, the Cafe is not doubtful when we undertake to find the Latitude of Rome; unless it be not known whether it lies to the Northward or Southward of London; which however could not be determined by any Principies of Trigonometry. Likewise, in Astronomy, if the Latitude of the Place, the Sun's Declination and Azimuth, were given, the Quæfitum is not doubtful neither, unJess the Sun's Declination exceeds the Latitude of the Place, and both are of the fame Denomination, that is, both North or both South; in which Cases, because it is possible for the Sun to be upon the same Azimuth Circle, twice in the Forenoon, and upon another Azia muth Circle, twice in the Afternoon; it is doubtsul, if by Circumstances, during the Observation, we can't dilcover which of the Times, whether the first, or last; but if those Times fall near each other, it will be quite impoflible to distinguish which, and therefore ambi. guous. Other Instances might be produced, but I believe these are sufficient to evince, that those nice Diltinctions are not fo necessary in Practice : If there be those who think otherwise, I shall not dispute it, but leave them to their Opinion without Interruption.
However, what with Mr. Cunn's Rules for determining the ambiguous Cases (which are judiciously drawn up, as including all the Varieties poflible), and the Corrections now made by restoring what was loft and corrupted, our Author's Treatise of Trigonometry, in respect to Theory, may perhaps appear compleat, even to the most scrupulous. And,
Here I thought to conclude ; but, for the Sake of Novelty, and to illustrate the various Methods for solv