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The seventh Case much resembles the third ; for
there are given two Sides B C, CD, and B an
Angle, opposite to CD one of them; to find the Angle BCD, lying between those two Sides. ND here we may observe, that the Species of the
S the same 2 Angle DCA is known; for it is of
S less Kind with the Angle B, when DC is
2 than a Quadrant. And,
If DCA be less than BCA, and the Sum of DCA and BCA less than 2 right Angles; then, DCA either added to, or subtracted from BCA will give the Angle BCD; which, therefore, is ambiguous.
If DCA be not less than BCA, or the Sum of DCA and BCA not less than two right Angles; then their Sum in the former, and their Difference in the latter Variety shall give the single Value of BCD; which, then, is not ambiguous.
N. B. If any one will be at the Trouble to make a
double Calculation for the Side DC, or the Angle D, as taught in the Remarks on the 9th and 10th Cases, they will find the several Varieties in the if, 3d, 5th, and 7th, to be as here laid down in these easy Rules.
The Truth of these Rules may be easily deduced from
the 10th, 13th, 18th, and 22d Prop. of this, and the 2d, 8th, and 13th Examples, following Prop.
30. of this.
In our third Case of oblique-plain Triangles, our Au
thor should have added this.
If A B be less than BC, the Angle A is ambiguous. otherwise note
TRE A TI SE
Nature and Arithmetick
HE Mathematicks formerly received considerable Advantages ; first, by the Introduction of the Indian Charaĉters, and
afterwards by the Invention of Decimal Fractions ; yet bas it since reaped at least as much from the Invention of Logarithms, as from both the other two. The Use of these, every one knows, is of the greatest Extent, and
runs through all Parts of Mathematicks. By their Means it is that Num. bers almost infinite, and such as are otherwise impracticable, are managed with Ease and Expedie tion. By their Asistance the Mariner steers his Vessel, the Geometrician investigates the Nature of the big ber Curves, the Astronomer determines the Places of the Stars, the Philosopher accounts for.o.
ther Phenomena of Nature ; and lastly, the Ufuo rer computes the Interest of his Money.
The Subject of the following Treatise has been cultivated by Mathematicians of the first Rank ; some of whom taking in the whole Doctrine, bave indeed wrote learnedly, but scarcely intelligible to any but Masters. Others, again, accommodating themselves to the Apprehension of Novices, bave selected out some of the most easy and obvious Properties of Logarithms, but have left their Nature and more intimate Properties untouch'd. My Design therefore in the following Tract, is to supply what seem'd still wanting, viz. to discover and explain the Doctrine of Logarithms, to those who are not yet got beyond the Elements of Algebra and Geometry.
The wonderful Invention of Logarithms we owe to the Lord Neper, who was the first that conAtructed and published a Canon thereof, at Edinburgh, in the Year 1614. This was very graciously received by all Mathematicians, who were immediately sensible of the extreme Usefulness thereof
. And tho’ it is usual to have various Nations contending for the Glory of any notable Invention, yet Neper is universally allowed tbe Inventor of Logarithms, and enjoys the whole Honour thereof witbout any Rival.
The fame Lord Neper, afterwards invented another and more commodious Form of Logarithms, which he communicated to Mr. Henry Briggs, Professor of Geometry at Oxford, who was herebyrintroduced as a Sbarer in the compleating thereof: But the Lord Neper dying, the whole Business remaining, was devolved upon Mr. Briggs, who, with prodigious Application, and an uncommon Dexterity, compos'd a Logarithmick Canon, agreeable to that new Form for the first twenty Chiliads
of Numbers, for from to 20000) and for eleven other Chiliads, viz, fragma 20000 to J91900. For all wbich Numbers be calculated the Logarithmas to fourteen Places of Figures. This canin was publisk'd at London in the Year 1624,
Adrian Vlacq published again this Canon at Gouda in Hohand, in the Year 1628, with tke intermediate Chiliads before omitted, filled'te 12 de cording to Briggs's Prescriptions ; but these Tables are not so useful as Briggs's, because the Logarithms are continued but to 10 Places of Figures.
Mr. Briggs also has calculated the Logarithms of the Sines and Tangents of every Degree, and ibe bundredth Parts of Degrees to 15 Places of Figures, and has subjoined to them the natural Sines, Tangents, and Secants, to 15 Places of Figures. The Logarithms of the Sines and Tangents are called artificial Sines and Tangents. These Tables, together with their Construktion and use, was publisd after Briggs's Death, at London, in the Year 1633, by Henry Gelibrand, and by him called Trigonometria Britannica.
Since then there bave been published, in several Places, compendious Tables, wherein the Sines and Tangents, and their Logarithms, consist of but seven Places of Figures, and wherein are only the Logarithms of the Numbers from 1 to 100000, which may be sufficient for most Uses.
The best Disposition of these Tables in my Opinion, is that first thought of by Nathaniel Roe, of Suffolk ; and with some Alterations for the better, followed by Sherwin in bis Mathematical Tables, published at London in 1705 ; wherein are the Logarithms from 1 to 101000, confifting of 7 Places of Figures. To which are subjoined Y 2
the Differences and proportional Parts, by Means of which may be found easily the Logarithms of Numbers to 10000000, observing at the same Time that tbese Logarithms consist only of 7 Places of Figures. Here are also the Sines, Tangents, and Secants, with the Logarithms and Differences for every Degree and Minute of the Quadrant, with some other Tables of Use in practical Matbematicks,