which thence is ambiguous. And when only one of thefe Sums is lefs than a Semicircle, the acute Value of D A, added to B A, 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 D A is lefs than BA, BD will be had by fubtracting either Value of D A from BA; and then B D is ambiguous. But when the obtufe Value of D A is not lefs than B A, 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 Omiffion; where there are given two Sides B C, CD, and B an Angle oppofite to CD one of them, to find the third Side B D. FIRST, we may obferve, that the Species of DA is S the fame always known; for it is of a different Affection with the Angle B, when DC is {lefeater} drant. And, than a Quan If A D be less than A B, and also the Sum of AD and A B lefs than a Semicircle; then AD either added to, or fubtracted from A B, will give the Value of BD; which, therefore, is ambiguous. But if A D be not lefs than AB, or if their Sum be not less than a Semicircle; then their Sum in the former, and their Difference in the latter Variety, fhall give one fingle Value of BD; and then it is not ambiguous. The feventh Cafe much resembles the third; for ANI Kind with the Angle B, when DC is les a Quadrant. And, { greater } than If DCA be lefs than BCA, and the Sum of DCA and B C A lefs than two Right Angles; then DCA, either added to, or fubtracted from BCA, will give the Angle BCD; which, therefore, is ambiguous. If D C A be not lefs than B C A, 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, 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 of 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 thefe eafy 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 plane 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 Arithmetic OF LOGARITHMS. Th The PREFACE. HE Mathematics formerly received confiderable Advantages; firft, 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 Mathematics. By their Means it is that Numbers almost infinite, and fuch as are otherwise impracticable, are managed with Eafe and Expedition. By their Affiftance the Mariner steers his Veffel, the Geometrician investigates the Nature of the higher Curves, the Aftronomer determines the Places of the Stars, the Philofopher accounts for Y 3 other other Phanomeua of Nature; and, laftly, theUfurer 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 written learnedly, but fcarcely intelligibly 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, untouched. My Defign therefore in the following Tract is, to Jupply what feemed still wanting, viz. to discover and explain the Doctrine of Logarithms, to thofe 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 conftructed and published a Canon thereof, at Edin burgh, in the Year 1614. This was very graciously received by all Mathematicians, who were immediately fenfible of the extreme Usefulness thereof. And though 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 |