1 — I—x” = L; and I-x==L2 = 1-1 L + { n2 L2 — n3 L3 + ÷ n2 L+, &c.

Whence I +x=1+nL+ { n2 L2 + ž n3 L3,

&c.

is a general Theorem for finding the Number from the Logarithm given of any Species or Form whatfoever; but in the Application of it to Practice we labour under a great Inconvenience, especially if the Numbers are large; that is to fay, it converges fo very. flow, that it were much to be wifhed it could be contracted.

However, if L be the Logarithm of the Ratio of a the leffer Term, to b the greater, and either of them are given; then the other will be easily had, and expeditiously enough too :

b a

For or

a

ГЂ

b

= 1 + n L + { n2 L2 + j, n3 L3, &c.

Wherefore it follows, by the Help of a Table of Logarithms, that the correfponding Number to any Logarithm may be found, to as many Places of Figures as thofe Logarithms confift of: For, putting d equal to the Difference between the given Logarithm and the next less in the Table, then will the Number fought, viz. Na × 1 + nd + 1⁄2 n2 d2 + 1⁄2 n3 d3, &c. But if d be put equal to the Difference between the given Logarithm, and the next greater, then NbxInd + { n2 d2 — n3 d3, &c. both which Series converge tafter, as d is smaller.

But the first three Terms in each may be contracted into two, which is very ufeful, inafmuch as it faves the Trouble of raifing n and d in the third Term to the fecond Power: For letting the firft Term remain as it is, the other two are reduced to one, thus ; make the fecond Term the Numerator of a Fraction, and Unity minus the third Term divided by the fecond is the Denominator.

Whence Na × 1 + nd + 1⁄2 n2 d'

becomes Na+

ad

;

2

d-d

2

m+d

will be the

Number anfwering to the given Logarithm; which, tho' it differs a little from the Truth, is fufficient to find the Numbers, exact to as many Places as Briggs's Logarithms confift of, viz. 14, which are the largest Tables extant. Much after the fame Method may the whole Series be contracted; by which Means each alternate Power of d will be exterminated; or, which is the fame Thing, every two Terms in the Series will be reduced to one, making the Whole shorter by Half.

To illuftrate these Conftructions by an Example: Let it be required to find the Number answering to the Logarithm 7,5713740282 in Briggs's Form. From the given Logarithm 7,5713740282 Subtract the Log. of 372710000 the 7,5713710453

next nearest

The Remainder is equal to d=

,0000029829

And because the Number 372710000 is lefs than the Number fought, call it a, which, multiplied by ,0000029829, and the Product 1,111756659, &c. divided by md,4342929, &c. quotes 2559,92; which, added to 372710000, gives 372712559,92 for the Number fought.

Thus, I prefume, the Doctrine of Logarithms has been fufficiently exemplified, whether we confider the Conftruction of them for any given Numbers; or, on the contrary, the finding of the Numbers from the Logarithms given.

But, before I conclude, I fhall give an Inftance or two of the great Ufe of Logarithms in Arithmetical Calculations, and firft in the purchafing of Annuities.

If a be put for any Annuity, p for the prefent Value, r the Amount of One Pound for One Year at any Rate of Intereft, and t for the Time or Number of

Years the Annuity is to continue; then p=

a

rt

r

a

the Value of the Annuity.

Cc 2

EX

EXAMPLE.

Let it be required to find the prefent Value of an Annuity of 60 l. per Annum, to continue 75 Years, at the Rate of 4 per Cent. per Annum.

Here a=60, t=75, and r=1,04. Now, in order to obtain the Anfwer, we muft find the feventy-fifth Power of r, or of 1,04; that is, we must multiply 1,04 feventy-five Times into itself, which is exceeding tedious by the comnion Way, as any one may judge; but by the Logarithms 'tis done with the greatest Ease; for if 0,0170333 the Logarithm of 1,04 be multiplied by 75, the Product 1,2774975 will be the Logarithm of the feventy-fifth Power of 1,04; which being fubtracted from 1,7781512, the Logarithm of a equal to 60, will leave 0,5006537 the Logarithm of 3,167041, which being fubtracted from 60, and the Remainder divided by r-1,04 will give 1420,824 = 1420l. 16 s. 53d. for the Value of the Annuity; and if 1420,824 be divided by 60, the Quotient will exhibit the Number of Years Purchafe requifite to be given for any Annuity to continue 75 Years upon a good Security free of all Incumbrances, the Purchase being made at 4 per Cent.

Hence we fee the Reafon why the long Annuities purchased in the Year 1708, having about 75 Years to come, are valued in Caftain's Bill of Exchanges at 24 or 25 Years Purchase: For, tho' according to this Calculation, they are worth but a little more than 23. Years and a Half; yet, becaufe in the public Funds 4 per Cent. is fcarcely ever made of Money, and the Contingencies it is there fubject to, which thofe Annui-" ties, and other Government Securities, are not, makes them very justly worth 24 or 25 Years Purchase.

Likewife Queftions relating to Annuities upon Lives, whether for one, two, or three, &c. are almost as eafily eftimated. For Inftance; it may readily be found by Logarithms, that an Annuity for a Man of Thirty, to continue during his Life, is worth 11,61 Years Purchase, Intereft 6 per Cent. but at 4 per Cent. 14,68. And as the Probabilities of Life's Continuance, and the Value thereof, are determined by an algebraical Procefs grounded upon the Rudiments of the Doctrine of Chances, and five Years Obfervations upon the Bills of Mortality of Breslaw, the Capital of Silefia; fo there refults that Truth and

Equity from the Operations, as ought to prefide in all Contracts of this Nature. Whence it follows, that all other Methods, whofe Refolution differs from this (especially if the Difference be much), may justly be deemed erroneous, and confequently prejudicial to one of the Parties concerned. Wherefore, to prevent Impofitions thro' Ignorance, great Care fhould be taken; which Precaution, however unneceflary it may appear, 'tis prefumed, it will be regarded, inasmuch as no one is willing to pay more Years Purchase than he has Chances for living; as, on the contrary, the Seller to receive less than his Due; which may poffibly be by following the common Methods (where, for the most part, Regard is had neither to Age nor Intereft) founded upon Caprice, Humour, or, if you pleafe, Cuftom, the Contract being made, as they can agree, right or wrong; which Method of Procedure ought to be exploded, fince fo liable to Error, and the Confequences drawn therefrom so often wide of the Truth.

The other Inftance which I fhall give of the great Ufe of Logarithms is in the Cafe of Seffa, as related by Dr. Wallis in his Opus Arithmeticum from Alfephad (an Arabian Writer), in his Commentaries upon Tograius's Verfes; namely, that one Seffa, an Indian, having first found out the Game at Cheffe, and fhewed it to his Prince Shehram, the King, who was highly pleased with it, bid him afk what he would for the Reward of his Invention; whereupon he afked, That for the firft little Square of the Chesse Board, he might have one Grain of Wheat given, for the second two, and fo on, doubling continually, according to the Number of the Squares in the Cheffe Board, which was 64. And when the King, who intended to give a very noble Reward, was much difpleafed, that he had afked fo trifling an one, Sefa declared, that he would be contented with this fmall one. So the Reward he had fixed upon was ordered to be given him. But the King was quickly aftonifh'd, when he found, that this would rife to fo vatt a Quantity, that the whole Earth itself could not furnish out so much Wheat. But how great the Number of thefe Grains is, may be found by doubling one continually 63 Times, fo that we may get the Number that comes in the laft Place, and then one Time more to have the Sum of all; for the Double of the last Term lefs by

Cc 3

by one is the Sum of all. Now this will be most expeditiously done by Logarithms, and accurately enough too for this Purpofe: For if to the Logarithm of 1, which is o, we add the Logarithm of 2 (which is 0,3010330) multiplied by 64, that is, 19,2659200, the abfolute Number agreeing to this will be greater than 18446, 00000, 00000, 00000, and less than 18447, 00000, 00000, 00000.

As I have had the revifing of these Sheets, fo it may be expected that I fhould give my Opinion concerning Mr. Cunn and our Author, in regard to Spherical Trigonometry; wherein the former accufes the latter, and feveral other eminent Authors, of having committed many Faults, and, in fome Cafes, of being mistaken, efpecially in the Solution of the 12th Cafe of Oblique Spherics; in which Mr. Cunn has intirely mistaken the Author's Meaning, as plainly appears by his Remark, where he constitutes a Triangle whofe Sides are equal to the given Angles; whereas the Author means, that each Angle fhould first be changed into its Supplement, and then with the faid Supplements another Triangle conftituted, whofe Angles, by the very_Text of the 14th Propofition of his own Spherical Trigonometry, will be the Supplements of the Sides fought in the given Triangle; to which Propofition I refer the Reader. That this is the Senfe of the Author, is very evident, if impartially attended to, and which I think could poffibly have no other Meaning; and accordingly aver what is here advanced to be univerfally true; but, because I would not be misunderstood, fhall illuftrate the Truth thereof by a numerical Operation; which, to those who care not to trouble themselves with the Demonftration, may be sufficient; and, to others, fome Satisfaction.

EXAMPLE.

Suppofe, in the Oblique-angled Spherical Triangle ADE, there are given the Angles A, D, E, as per Figure, and the Side DE required.

Note, Write down the Supplements of the two Angles next the Side required firft; and then the Operation may ftand thus:

The