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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 otherwife, is to be ungenerous; becaufe, whenever thofe Miftakes happen, as they are for the moft part owing more to Inadvertency, than Want of Knowledge; fo they fhould therefore be attributed to the Frailty of human Nature (to which we are all more or lefs fubject), 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 justify'd Mr. Cunn: For fuch an Occafion as this offering, I thought the Dif ference 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 ofall the dif ferent 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 moft part, as well, or, at leaft, with more Eafe, from the Conftruction 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 stop, for want of farther Explications, but with more Eafe fupply 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 Perfon will have but a mean Notion of this useful Branch of the Mathematics); and, if fo, he ought in fome measure to be excufed, efpecially 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; befides what is ambiguous in Trigonometry, is very often not fo in Geography and Aftronomy, &c. for which the other is chiefy learnt.
For Inftance: 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 oppofite to one of them, given, the Cafe is not doubtful when we undertake to find the Latitude of Rome; unlefs 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. Likewife, in Aftronomy, if the Latitude of the Place, the Sun's Declination and Azimuth, were given, the Quæfitum is not doubtful neither, unJefs 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 Cafes, because it is poffible for the Sun to be upon the fame Azimuth Circle, twice in the Forenoon, and upon another Azimuth Circle, twice in the Afternoon; it is doubtful, if by Circumftances, during the Obfervation, we can't dif cover which of the Times, whether the firft, or last; but if thofe Times fall near each other, it will be quite impoffible to diftinguifh which, and therefore ambiguous. Other Inftances might be produced, but I believe these are fufficient to evince, that thofe nice Diftinctions are not fo neceflary in Practice: If there be those who think otherwife, I fhall not difpute it, but leave them to their Opinion without Interruption.
However, what with Mr. Cunn's Rules for determining the ambiguous Cafes (which are judiciously drawn up, as including all the Varieties poffible), and the Corrections now made by reftoring what was loft and corrupted, our Author's Treatife of Trigonometry, in respect to Theory, may perhaps appear compleat, even to the moft fcrupulous. And,
Here I thought to conclude; but, for the Sake of Novelty, and to illuftrate the various Methods for folv
ing the 12th Cafe of Oblique Spherics, where the three Angles are given to find either of the Sides, I fhall beg Leave to give one Inftance more, in order to fhew how it may be perform'd after a new Manner, by the Help of the natural and logarithmetic Verfed Sines; which, if not intirely new, is not fo publicly known as the preceding Methods; at least, I never faw any-where the Method of Operation, and therefore fhall deliver a Rule for that Purpose, in the following Words:
Having according to the former Directions, chang'd one of the Angles next the Side fought into its Supplement; take the natural Verfed Sine of the Difference of the faid Supplement and the other adjacent Angle, and fubtract it from the natural Verfed Sine of the Angle oppofite to the Side fought, and to the Logarithm of the Remainder add the Square of the Radius; then from the Sum fubtract the logarithmetic Sines of the above Supplement, and the fame adjacent Angle; and the Remainder is the Logarithm of a Number, which will be the Verfed Sine of the Side fought.
Which Remainder 9,656450 gives the Logarithm Verfed Sine of DE 56° 42', agreeing exactly with the former Computations.
Note, If the faid Remainder exceeds 10,000000, it implies that the Side fought is greater than a Quadrant; wherefore cancelling the Characteristic ro, look out for the Number anfwering the remaining Logarithm,
from which cut off the Left-hand Figure, or, which is the fame Thing, abate the Radius (viz. Unity); and the Remainder will be the natural Sine of the Excefs of the Side fought above a Quadrant.
As the natural and logarithmetic Verfed Sines are not fo frequently met with in Books as the artificial Sines, 'tis poffible, on that Account, this Rule may meet with fome Objection; for which Reafon, and not knowing whether it may be thought preferable to the foregoing Methods (tho' undoubtedly very eafy in Practice), I have omited its Demonstration; but have publifhed the Rule, with some View of introducing the Use of the former Sines, which fometimes are preferable to the latter For by the Help of the faid Verfed Sines, and the Reasoning used in obtaining this Rule, we neceffarily come to the Knowledge of folving that Problem, where two Sides and the contained Angle are given, and the third Side required, at one Operation, very ufeful in Aftronomy and Geography, especially in the latter; when the Latitudes and Longitudes of two. Places are given to find their Diftance afunder: But the Rule for performing it, and the Demonftration thereof, is alfo omitted for the Sake of Brévity.
However, 'tis eafy to perceive, fince Angles may be turned into Sides, that the present Rule includes the Solution of that ufeful Problem in Aftronomy for finding the Sun's Azimuth, having the Latitude of the Place, the Sun's Altitude and Distance from the elevated Pole given; by which means the Variation of the Compafs, of fuch Importance to Navigators, may be readily determined in any Part of the World.
An Example of which, comprehending the latter Part of the Rule (viz. when the Remainder exceeds 10,000000) is exhibited.
Suppofe on June the 30th, 1732, at London, in the Latitude of 51° 52' N. it were required to find the Sun's true Azimuth, when his Altitude was 50° 00', in the Afternoon. First,
Here the Remainder exceeds 10,000000; wherefore cancel the Characteristic 10, and the Number answering the remaining Logarithm is 1,5591; the Excess of which above Unity, viz.,5591, gives the natural Sine of 34° 00'; whence the Sun's true Azimuth is North 124° 00' Weft: At which Time, if the Sun's Magnetical Azimuth were North 110° 30' Weft, the Variation of the Compafs would be 13° 30' Weft, as appears by the following Subtraction.
True Azimuth North, 124° 00' Weft
Variation 13 30 Weft
N. B. If the Sun's Declination had been South, then the Verfed Sine of the Sun's Distance from the elevated Pole would have been equal to Unity plus the natural Sine of the Sun's Declination; which in Practice creates no more Trouble than when the Declination is North, if fo much; fince it is at leaft as easy to take the natural Sine of an Arc, as to take the Verfed Sine of its Complement to go Degrees; which Sines, and others, with their refpective Logarithms, c. may readily be had out of Sherwin's Mathematical Tables.