metical complement, together with its abfolute number; to this tabular logarithm add the logarithm that was given, and the fum will be a logarithm neceffarily falling among those near the end of the ta-ble; find then its abfolute number, corrected by means of the proportional part, which will not be very inaccurate, as falling near the end of the table; this being divided by the abfolute number, before found for the logarithm next lefs than the arithmetical complement, the quotient will be the required number anfwering to the given logarithm; which will be much more correct than if it had been found from the proportional part of the difference where it naturally happened to fall and the reafon of this operation is evident from the nature of logarithms. But as this divifor, when taken as the number answering to the logarithm next lefs than the arithmetical complement, may happen to be a large prime number; it is farther remarked, that inftead of this number and its logarithm, we may use the next lefs compofite number which has small factors, and its logarithm; because the divifion by thofe fmall factors, instead of by the number itself, will be performed by the fhort and easy way of divifion in one line. And for the more eafy finding proper compofite numbers and their factors, our author here fubjoins an abacus or lift of all fuch numbers, with their logarithms and component factors, from 1000 to 10000; from which the proper logarithms and factors are immediately obtained by infpection. Thus, for example, to find the root of 10800, or the mean proportional between 1 and 10800:" The logarithm of 10800 is 4,03342,37554,8695, the half of which is 2,01671,18777,4347 the logarithm of the number fought, the arithmetical complement of which logarithm is o 98328,81222,5653; now the nearest logarithm to this in the abacus is 0,98227,12330,3957, and its annexed number is 9600, the factors of which are 2, 6, 8; to this laft logarithm adding the logarithm of the number fought, the fum is 0,99898,31107,8304, whofe abfolute number, corrected by the proportional part, is 99766,12651,6521, which being divided continually by 2, 6, 8, the factors of 96, the laft quotient is

103,92 304845471; which is pretty correct, the true number being' 103.923048454133=10800.

We now arrive at the 12th and 13th chapters, in which our ingenious author firft of all teaches the rules of the Differential Method, in conftructing logarithms by interpolation from differences. This is the fame method which has fince been more largely treated of by later authors, and particularly by the learned Mr. Cotes in his Canonotechnia. How Mr. Briggs came by it, does not well appear, as he only delivers the rules, without laying down the principles or inveftigation of them. He divides the method into two cafes, namely when the fecond differences are equal or nearly equal, and when the differences run out to any length whatever.

The former of these is treated in the 12th chapter; and he particularly adapts it to the interpolating 9 equidiftant means between two given terms, evidently for this reafon, that then the powers of 10 become the principal multipliers or divifors, and fo the operations performed mentally. The fubftance of his procefs is this: Having given two abfolute numbers

I 45

2 35

25

with their logarithms, to find the logarithms of 9 arithmetical means between the given numbers: Between the given logarithms take the Ift difference, as well as between each of them and their next or equidiftant greater and lefs logarithms; and likewife the 2d differences, or the two differences of these three Ift differences; then if thefe 2d differences be equal, multiply one of them feverally by the numbers 45, 35, &c, in the annexed tablet, dividing each product by 1000, that is cutting off three figures from each; laftly 8 25 to of the 1ft difference of the given logarithms

345678

15

5

555

15

9 35

Subductive

products.

Additive

products.

add feverally the first five quotients, and fubtract 10 45 the other five, fo fhall the ten refults be the refpective ift differences to be continually added, to compofe the required feries of logarithms. Now this amounts to the fame thing as what is at this day taught in the like cafe we know that if A be any term of an equidiftant feries of terms, and a, b, c, &c, the first of the 1ft, 2d, 3d, &c, o of Order differences; then the term z, whofe diftance from A is expreffed by *, will be thus, z=A+xa + x.*—1 b + x.

2

X-I x-2 +

2

3

C &c. And:

if now, with our author, we make the 2d differences equal, then c, d, e, &c, will all vanish or be equal to O, and z will become barely

form, as he thought it easier to multiply by 5 than to divide by 2. Also all the last terms (x.—b) are set down positive, because in the logarithms b is negative.--If the two 2d differences be only nearly equal, take an arithmetical mean between them, and proceed with it the fame as above with one of the equal 2d differences.-He alfo fhews how to find any one fingle term, independent of the reft and concludes the chapter with pointing out a method of finding the proportional part more accurately than before.

In the 13th chapter our author remarks, that the best way of filling up the intermediate chiliads of his table, namely from 20000 to 90000, is by quinquifection, or interpofing four equidiftant means between two given terms; the method of performing which he thus particularly defcribes. Of the given terms, or logarithms, and two or three others on each fide of them, take the 1ft, 2d, 3d, &c differences,

till the laft differences come out equal, which fuppofe to be the 5th differences: divide the 1ft differences by 5, the 2d by 25, the 3d by 125, the 4th by 625, and the 5th by 3125, and call the refpective quotients the ift, 2d, 3d, 4th, 5th mean differences; or, instead of dividing by these powers of 5, multiply by their reciprocals, TOOS, TO 650, TO3 that is multiplied by 2, 4, 8, 16, 32, cutting off refpectively one, two, three, four, five figures from the end of the products, for the feveral mean differences: then the 4th and 5th of thefe mean differences are fufficiently accurate, but the 1ft, 2d, and 3d are to be corrected in this manner; from the mean third differences fubtract three times the 5th difference, and the remainders are the correct 3d differences; from the mean 2d differences subtract double the 4th differences, and the remainders are the correct 2d differences; laftly from the mean 1ft differences take the correct 3d differences, and of the 5th difference, and the remainders will be the correct firft differences. Such are the corrections when the differences extend as far as the 5th. However in compleating those chiliads in this way, there will be only 3 orders of differences, as neither the 4th nor 5th will enter the calculation, but will vanifh through their fmallness : therefore the mean 2d and 3d difference will need no correction, and the mean first differences will be corrected by barely fubtracting the 3d from them. Thefe preparatory numbers being thus found, all the 2d differences of the logarithms required, will be generated by adding continually, from the lefs to the greater, the conftant 3d difference; and the feries of 1ft differences will be found by adding the feveral 2d differences; and laftly by adding continually these ift differences to the ft given logarithm &c, the required logarithmic terms will be generated.

Thefe eafy rules being laid down, Mr. Briggs next teaches how by them the remaining chiliads may beft be compleated: namely, having here the logarithm for all numbers up to 20000, find the logarithm to every 5 beyond this, or of 20005, 20010, 20015, &c, in this manner; to the logarithms of the 5th part of each of those, namely 4001, 4002, 4003, &c, add the conftant logarithm of 5, and the fums will be the logarithms of all the terms of the feries 20005, 20010,20015, &c: and thefe logarithms will have the very fame differences as thofe of the feries 4001, 4002, 4003, &c; by means of which therefore interpofe 4 equidiftant terms by the rules above; and thus the whole canon will be eafily compleated.

He here alfo extends the rules for correcting the mean differences in quinquifection, as far as the 20th difference; he alfo lays down fimilar rules for trifection, and speaks of general rules for any other fection, but omitted as being lefs eafy. So that he appears to have been poffeffed of all that. Cotes afterwards delivered in his Cano→ notechnia five Conftructio Tabularum per Differentias, drawn from the Differential Method, as their general rules exactly agree, Briggs's mean and correct differences being by Cotes called round and quadrat differences, because he expreffes them by the numbers 1, 2, 3, &c, written respectively in a small circle and square.

Mr. Briggs also observes that the fame rules equally apply to the conftruction of equidiftant terms of any other kind, fuch as fines, tangents, fecants, the powers of numbers, &c: and farther remarks, that of the fines of three equidifferent arcs, all the remote differences may be found by the rule of proportion, because the fines and their 2d, 4th, 6th, 8th, &c differences are continued proportionals, as are also the Ift, 3d, 5th, 7th, &c differences among themfelves; and like as the 2d, 4th, 6th, &c differences are proportional to the fines of the mean arcs, fo alfo are the 1ft, 3d, 5th, &c differences proportional to the cofines of the fame arcs. Moreover with regard to the powers of numbers, he remarks the following curious properties; 1ft, that they will each have as many orders of differences as are denoted by the index of the power, the fquares having two orders of differences, the cubes three, the 4th powers four, &c: fecond, that the laft differences will be all equal, and each equal to the common difference of the fides or roots raised to the given power and multiplied by 1 × 2 × 3 × 4 &c, continued to as many terms as there are units in the index; fo if the roots differ by 1, the 2d difference of the fquares will be each 1 x 2 or 2, the 3d differences of the cubes each i x 2 x 3 or 6, the 4th differences of the 4th powers each 1 × 2 × 3 × 4 or 24, and fo on; and if the common difference of the roots be any other number n, then the laft differences of the fquares, cubes, 4th powers, 5th powers, &c, will be refpectively 2n2, 6n3, 24×4, 120n5, &c.

Befides what was fhewn in the eleventh chapter concerning the taking out the logarithms of large numbers by means of proportional parts, he employs the next or 14th chapter in teaching how, from the first ten chiliads only, and a fmall table of one page, here given, to find the number anfwering to any logarithm, and the logarithm to any number confifting of fourteen places of figures.

Having thus fully fhewn the conftruction and chief properties of his logarithms, our ingenious author, in the remaining eighteen chapters, exemplifies their ufes in various curious and important fubjects; fuch as The Rule-of-three, or rule of proportion; finding

*It is no more than a large exemplification of this method of Briggs's that has been printed fo late as 1771, in a 4to. tract by Mr. Rob. Flower, under the title of The Radix, a New Way of making Logarithms. Although Briggs's work might not be known to this writer. Since this was written 1 have been favoured with the following anecdote, concerning Mr. Flower and his work, by the Rev. Dr. Horfley, the learned editor of the works of Sir I. Newton. "This Robert Flower was a very obfcure, and probably an illiterate man. He was master of a writing school in the town of Bishop Stortford in Hertfordshire. He communicated his Radix, before he published it, to my late learned friend Math. Raper, Efq. of Thorley Hall. I was at Thorley at the time, upon a vifit to my father, who was rector of the parish; and I well remember that Mr. Raper told me with great furprize, that Flower (who was known to us both by name as the writing-mafter of the neighbouring market town) had fallen upon Briggs's way of finding all logarithms from the first ten chiliads. And he was fo well perfuaded that Flower had made the difcovery for himself, without any light from Briggs, that with his accustomed munificence he rewarded the man's ingenuity with a prefent of ten guineas; informing him I believe that his work had been done before, and diffuading the publication."

the roots of given numbers; finding any number of mean proportionals between two given terms; with other arithmetical rules: Alfo various geometrical fubjects, as ift, Having given the sides of any plane triangle, to find the area, perpendicular, angles, and diameters of the infcribed and circumfcribed circles; 2d, In a rightangled triangle, having given any two of thefe, to find the reft, viz. one leg and the hypotenufe, one leg and the fum or difference of the hypotenuse and the other leg, the two legs, one leg and the area, the area and the fum or difference of the legs, the hypotenufe and fum or difference of the legs, the hypotenufe and area, and the perimeter and area; 3d, Upon a given bafe to defcribe a triangle equal and ifoperimetrical to another triangle given; 4th, To defcribe the circumference of a circle fo, that the three diftances from any point in it to the three angles of a given plane triangle, fhall be to one another in a given ratio; 5th, Having given the bafe, the area, and the ratio of the two fides of a plane triangle, to find the fides; 6th, Given the bafe, difference of the fides, and area of a triangle, to find the fides; 7th, To find a triangle whofe area and perimeter fhall be expreffed by the fame number; 8th, Of four given lines, of which the fum of any three is greater than the fourth, to form a quadrilateral figure about which a circle may be defcribed; 9th, Of the diameter, circumference, and area of a circle, and the furface and folidity of the fphere generated by it, having any one given, to find any of the reft; 10th, Concerning the ellipfe, fpheroid, and gauging; 11th, To cut a line or a number in extreme and mean ratio; 12th, Given the diameter of a circle, to find the fides and areas of the inscribed and circumfcribed regular figures of 3, 4, 5, 6, 8, 10, 12, and fixteen fides; 13th, Concerning the regular figures of 7, 9, 15, 24, and 30 fides; 14th, Of ifoperimetrical regular figures; 15th, Of equal regular figures; and 16th, Of the fphere and the 5 regular bodies; which closes this introduction. Such of these problems as can admit of it, are determined by elegant geometrical constructions, and they are all illuftrated by accurate arithmetical calculations performed by logarithms; for the exemplification of which they are purposely given.

At the end he remarks, that the chief and moft neceffary use of logarithms, is in the doctrine of spherical trigonometry, which he here promifes to give in a future work, and which was accomplished in his Trigonometria Britannica, to the defcription of which we now proceed.

Of BRIGGS's Trigonometria Britannica.

At the close of the account of writings on the natural fines, tangents, and fecants, I omitted the defcription of this work of our learned author, although it is perhaps the greatest of this kind, all things confidered, that ever was executed by one perfon; purposely referving my account of it to this place, not only as it is connected with the invention and construction of logarithms, but thinking it