your point of feparation, and prefixing cyphers where neceffary.'

With regard to the investigation of these factors, we are left entirely in the dark at prefent, but with a promise that a time may come, when the whole mystery fhall be revealed to us: take it in Mr. Wefton's own words: It may perhaps be expected I should inform the world by what method this species of figures was first discovered; but that is referved for a future publication.' We take no fort of delight in thus exposing the foibles of mankind; far from it: we would with rather to conceal them from public view, but when an author can, in this manner, be ferious upon the most trifling affairs (as Mr. Wefton really is) he certainly fubjects himself to ridicule. Does not every smatterer in decimal arithmetic know, that if unity be divided by the number of component parts of a proposed integer, the quotient will be a common multiplier for reducing any number of those parts to a decimal of the whole; and will it not follow from hence that Tiz or,089289, &c. is the mul- . tiplier for finding the decimal of any number of pounds, an hundred weight being the integer; also that or,83333, &c. becomes the factor for finding the decimal of any number of ounces, a pound Troy being the integer; and laftly, that 313 or,0027397, &c. denotes the common multiplier for finding the decimal of any number or days, a year being the integer? Is there any new discovery in all this? Certainly not; on the contrary we are rather inclined to believe it very probable, that the method of finding thefe factors, fo far from being unknown is our forefathers, was well underflood even by their grandfathers.

We should not trouble our readers with any farther remarks upon this jejune performance, but there being fomething fo very curious in our author's manner of overcoming the diffi culties, frequently refulting from the application of these tabular factors to the folution of his own examples, we must beg leave to point out two or three of them, as a specimen of Mr. Wefton's skill in abbreviating decimal operations. Find the decimal of Lo: 17:9:2.

Page 7, example 4.

By the old method, the decimal required is ,820625. By the new method it is performed thus, viz.

17 fhillings

× by 5

Then, as mentioned in the general rule, one cypher being prefix'd to the product of pence, and two before that for farthings, you will find the decimal of this operation to be

for the fhillings

,03753

for the pence

,003126 for the farthings

,890656 decimal of 17 s. 9d. 3

20

17,813120

12

9,757440

4

3,029760 Proof.

'This proof does not come out quite fo near as the other, but as delicate calculations are not required in trade, the confequences are not material; and as it answers fufficiently for any calculations in bufinefs, by bringing out the exact sum; the excess in the remaining decimal is not worth obferving.' Page 22, line 14. I have abated 18 in this laft deduction, although but 17 tens' (for 17 hundreds) becaufe 8 is so near the abatement that it becomes neceffary; indeed, all general axioms must be also affifted with the reafon of the operator in 'trifles."

This may be our author's cafe, for aught we know, he having, as operator, certainly dealt in trifles, but whether he has affifted them with his reafon, we will not prefume to determine

VIII. Plain Trigonometry rendered eafy and familiar, by Calculations in Arithmetic only: with its Application and Ufe in afcertaining all kinds of Heights, Depths, and Distances, in the Heavens, as well as on the Earth and Seas; whether of Towers, Forts, Trees, Pyramids, Columns, Wells, Ships, Hills, Clouds, Thunder and Lightning, Atmosphere, Sun, Moon, Mountains in the Moon, Shadows of Earth and Moon, Beginning and End of Eclipfes, &c. In which is alfo fhewn, a curious Trigonometrical Method of difcovering the Places where Bees hive in large Woods, in order to obtain, more readily, the falutary Produce of those little Infects. By the Rev. Mr. Turner, late of Magdalen-Hall, Oxford. Folio. Pr. 2s. 6d.. Crowder.

more efpecially with regard to the doctrine of the sphere, the refolution

refolution of plain and spherical triangles, and feveral other parts of fpeculative and practical aftronomy; we must look upon every attempt towards facilitating the laborious operations by the tables of logarithms, fines, tangents, fecants, &c. as defigned for public utility: with this view our author has, from a due confideration of what has been already published (as he ingenuously owns in the addrefs to his readers) endeavoured to remove the trouble and difficulty attending the use of those tables, by fubftituting in their room, a fhort and easy method of operating in trigonometrical calculations, by common arithmetic only. In this Mr. Turner has fucceeded, as far as we are capable of judging, better than any preceding writer. His manner of defining the feveral fpecies of triangles, is very explicit and fatisfactory; and we muft add, that the investigations of the feveral cafes in plane trigonometry, deduced from a few general axioms, cannot fail of being very useful to the young tyro in mathematical ftudies.

As a fpecimen of our author's performance, we fhall lay before our readers the following extracts:

There are generally reckoned by writers on plane trigonometry (fays our author) feven cafes of right angled triangles; but by this method they are all reduced to four, the folutions of which depend on the following axioms:

AXIOM I. Divide 4 times the fquare of the complement of the angle, whofe oppofite fide is either given or fought, by 300 added to 3 times the faid complement; this quotient added to the faid angle, will give you an artificial number, called fometimes the natural radius*, which will ever bear the fame proportion to the hypothenufe, as that angle bears to its oppofite fide. In angles under 45 degrees, the artificial number may be found eafier thus: Divide 3 times the fquare of the angle itfelf, whofe oppofite fide is given or fought, by 1000; the quotient added to 57.3 †, a fixed number, that füm will be the artificial number required.-This is to be ufed, when the angles and a fide are given to find another fide.

AXIOM II. The fquare of both the legs, i. e. the fquare of the bafe and perpendicular added together, is equal to the fquare of the hypothenufe; whofe fquare root is the hypothe

=

The natural radius is only turning the right angle 90 degrees into an artificial number, which fhall always bear the fame proportion to the hypothenufe, as the given angle does to it oppofite leg..

57.3 is the radius of a circle whofe circumference is 360.

nufe

nufe itself. This is made ufe of, when the bafe and perpendicular are given to find the hypothenufe.

AXIOM III. The fum of the hypothenufe and one of the legs multiplied by their difference, the fquare root of that product will be the other leg required.-This comes into ufe, when the hypothenufe and one leg is given, to find the other leg.

AXIOM IV. Half the longer of the two legs, added to the hypothenufe, is always in proportion to 86 1, as the fhorter leg is to its oppofite, angle. -This is ufeful when the fides are given, to find the angles.

Note, Thefe 4 axioms will answer all the cafes of right and oblique angled triangles, except the laft cafe in obliques, which will require fome farther affiftance, and will be fhewn when we come to treat of that cafe.'

The fecond axiom is evidently an application of the 47th propofition of the first book of Euclid's Elements, and the third may be eafily deduced from it, for as the fum of any two quantities being multiplied by their difference, will produce the difference of the fquares of thofe quantities, it is therefore very obvious that the difference between the fquare of the hypothenuse and that of either leg (as it is equal to the fquare of the leg not taken, by the abovementioned propofition) must be equal to the product of the fum and difference of the hypothenufe and one of the legs. With regard to the first and fourth axioms, the principles are not fo evident from whence they were derived; however, as Mr. Turner has not given the procefs by which he obtained thofe ufeful approximations, we apprehend it will not be deemed impertinent in us just to mention the method of inveftigating thofe, or fimilar numbers for approximating, indefinitely near, the fine of any propofed

arch.

If radius be made equal to unity, the length of any arch, lefs than 90°. is nearly equal to one third of the difference between the chord of the arch itself, and eight times the chord of half that arch, whence by taking the halves of those chords, the fines of arches may be eafily found, the error being only part of the fifth power of the affigned arch, measured in parts of the radius.

1

Our author next proceeds to the folution of the several cafes of right and oblique angled plane triangles, wherein he has applied thefe arithmetical calculations, we think, with great propriety. In the remaining part of this work, Mr. Turner has

86 radius and half of a circle whofe circumference is 360.'

re

refolved, in an elegant manner, fome very curious and entertaining problems, particularly thefe following:

• Problem XV. To take the distance of the fun, moon, or any heavenly bodies.

Prob. XVII. To measure the height of a lunar mountain. Prob. XVIII. To measure the height of the atmosphere.

Prob. XXI. To calculate the diameter of the earth's shadow at the distance of the moon; and also, the diameter of the moon's fhadow of the earth.

Prob. XXII.. To calculate the beginning, end, and total duration of an eclipfe.

Prob. XXIV. To find, by a new method, where the bees bive in large and extenfive woods, in order to obtain their honey.'

Mr. Turner concludes this well conducted treatise thus:

• Thefe few problems are fufficient to point out the great ufe of this branch of learning. The advantages refulting from it to fociety are very great; almost infinite. Nothing however pofited in the heavens;-nothing upon the earth, or feas; -but its diftance and dimenfions may be ascertained by it.—— It is no wonder then, that Pythagoras, a learned philofopher of Samos, when he had discovered that famous propofition (47th of ift book of Euclid) which is the foundation of this fcience, fhould, in gratitude, facrifice an hecatomb, i. e. 100 oxen, to the mufes, for infpiring him with fuch an useful invention, which he judged beyond the power of human abilities to discover,

Thus by one plain geometrical figure, having three fides, and confequently as many angles, and affifted by the Rule of Three, you fee what amazing truths may be discovered. Tria funt omnia.'.

IX. A Letter to the reverend Vicar of Savoy to be left at J. J. Rouffeau's. Wherein Mr. Rouffeau's Emilius, or Treatise on Education, is humorously examined and exploded. Tranflated from the German of Mr. J. Mofer, Councellor of the High Court of Juftice at Ofnabruck, &c. &c. By J. A. F Warnecke, LL.C, a Native of Ofnabruck. 8vo. Pr. 15. DodЛley.

Umorously examined! Humour must be a very scarce commodity in Germany, if this pamphlet deferves that epithet there. Mr. Mofer addreffes himself to the reverend vicar, by whom we suppose he means Mr. Rouffeau; the sum of whofe doctrines is, that he acknowledges,a God, afferts conscience to be our judge, and admits of eternal punishments

and