Value of BD; which thence is ambiguous. And when only one of thefe Sums is less than a Semicircle, the acute Value of DA, added to BA, gives the only Value of BD; which then is not ambiguous; tho' in both Varieties the Perpendicular fell within.

2. When the Perpendicular falls without; that is, when the given Angles are of different Species.

WHEN the obtufe Value of DA is less than BA, BD will be had by fubtracting either Value of DA from BA; and then BD is ambiguous. But when the obtufe Value of DA is not less than BA, the acute Value of D A, taken from B A leaves the only Value of BD; which, therefore, is not ambiguous; tho' in both Varieties the Perpendicular fell

without.

In the third, we have the fame Omission; where there are given two Sides BC, CD, and B an Angle oppofite to CD one of them, to find the third Side BD.

IRST, we may obferve, that the Species of DA 2

F1

is always known; for it is of the fame Af

fection with the Angle B, when DC is

than a Quadrant. And,

lefs

greater}

If AD be less than A B, and alfo the Sum of AD and AB lefs than a Semicircle; then A D, either added to, or fubtracted from AB, will give the Value of BD, which, therefore, is ambiguous.

But if AD be not less than AB, or if their Sum be not lefs than a Semicircle; then their Sum in the former, and their Difference in the latter Variety, fhall give one fingle Value of B D, and then is not ambiguous.

The

The feventh Cafe much refembles the third; for
there are given two Sides BC, CD, and B an
Angle, oppofite to CD one of them; to find the
Angle BCD, lying between those two Sides.

AND here we may obferve, that the
Angle DCA is known; for it is of

Species of the the fame

Kind with the Angle B, when DC is than a Quadrant. And,

S lefs

2

greater}

If DCA be lefs than BCA, and the Sum of DCA and BCA less than 2 right Angles; then, DCA either added to, or fubtracted from B CA will give the Angle BCD; which, therefore, is ambiguous.

If DCA be not less than BCA, or the Sum of DCA and B C A not less than two right Angles; then their Sum in the former, and their Difference in the latter Variety fhall give the fingle 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 Cafes, they will find the feveral Varieties in the ift, 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 Cafe of oblique-plain Triangles, our Author fhould have added this.

If A B be less than B C, the Angle A is ambiguous. otherwise not,

A

A SHORT

TREATISE

OF THE

Nature and Arithmetick

O F

LOGARITHMS.

The PREFACE.

T

HE Mathematicks formerly received confiderable Advantages; first, by the Introduction of the Indian Characters, and afterwards by the Invention of Decimal Fractions; yet has it fince reaped at least as much from the Invention of Logarithms, as from both the other two. The Use of thefe, every one knows, is of the greatest Extent, and runs through all Parts of Mathematicks. By their Means it is that Numbers almost infinite, and fuch as are otherwife impracticable, are managed with Eafe and Expedition. By their Affiftance the Mariner steers bis Veffel, the Geometrician inveftigates the Nature of the bigber Curves, the Aftronomer determines the Places of the Stars, the Philofopher accounts for o-. Y

ther

ther Phenomena of Nature; and lastly, the Ufu rer computes the Intereft of his Money.

The Subject of the following Treatife has been cultivated by Mathematicians of the first Rank; fome of whom taking in the whole Doctrine, have indeed wrote learnedly, but fcarcely intelligible to any but Mafters. Others, again, accommodating themfelves to the Apprehenfion of Novices, have felected out fome of the most eafy and obvious Properties of Logarithms, but have left their Nature and more intimate Properties untouch'd. My Defign therefore in the following Tract, is to supply what seem'd ftill wanting, viz. to difcover 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 firft that conftructed and published a Canon thereof, at Edinburgh, in the Year 1614. This was very graciously received by all Mathematicians, who were immediately fenfible of the extreme Ufefulness thereof. And tho it is ufual to have various Nations contending for the Glory of any notable Invention, yet Neper is univerfally allowed the Inventor of Logarithms, and enjoys the whole Honour thereof without any Rival.

The fame Lord Neper afterwards invented another and more commodious Form of Logarithms, which he communicated to Mr. Henry Briggs, Profeffor of Geometry at Oxford, who was hereby introduced as a Sharer 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

of Numbers, (or from 1 to 20000) and for eleven other Chiliads, viz, from 90000 to 101000. For all which Numbers be calculated the Logarithms to fourteen Places of Figures. This Canon was publish'd at London in the Year 1624.

633

Adrian Vlacq published again this Canon at Gouda in Holland, in the Year 1628, with the intermediate Chiliads before omitted, filled up ascording to Briggs's Prefcriptions; but thefe Tables are not fo ufeful as Briggs's, because the Logarithms are continued but to 10 Places of Figures.

Mr. Briggs alfo bas calculated the Logarithms of the Sines and Tangents of every Degree, and the hundredth Parts of Degrees to 15 Places of Figures, and bas fubjoined 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. Thefe Tables, together with their Conftruction and Ufe, was publish'd after Briggs's Death, at London, in the Year 1633, by Henry Gelibrand, and by him called Trigonometria Britannica.

Since then there have been published, in feveral Places, compendious Tables, wherein the Sines and Tangents, and their Logarithms, confift of but feven Places of Figures, and wherein are only the Logarithms of the Numbers from 1 to 100000, which may be fufficient for moft Ufes.

The beft Difpofition of thefe Tables in my Opinion, is that first thought of by Nathaniel Roe, of Suffolk; and with fome Alterations for the better, followed by Sherwin in his Mathematical Tables, published at London in 1705; wherein are the Logarithms from 1 to 101000, confifting of 7 Places of Figures. To which are fubjoined

Y 2

the