For, as Columella remarks, (Lib. I. c. 4.) "it is certain that a large tract of land not rightly cultivated, will yield less than a fmaller space well cultivated."

The Account of Mr. Mills's Husbandry, to be concluded in our

next.

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, Clauds, 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 bive in large Woods, in order to obtain, more readily, the falutary Produce of thofe little Infects. By the Rev. Mr. Turner*, late of Magdalen-hall, Oxford. Folio. 2s. 6d. Crowder..

IN

Na fhort dedication of this little treatife, the Author observes, that the common method of anfwering trigonometrical problems being by large tables of fines, tangents and fecants, renders it not only expenfive by the purchase of them; but often precarious in the folution, by the mistakes of the prefs. I have therefore, adds he, for the ufe of the young mathematician, (from a confideration of what has been published on this curious fubject) compofed the prefent fyftem, by which any of the cafes in right or oblique plain triangles may be anfwered on the spot, by an eafy calculation in arithmetic only. great advantages refulting from this method to gentlemen in the army or navy, as well as to thofe in their private ftudies at home, muft immediately appear; as it will be found to anfwer the moft neceffary problems as expeditioufly as logarithms; and at the fame time wholly deliver you from thofe voluminous tables, and the inartificial fatigues of carrying them always with you.'

The

Having premifed these confiderations as a reason for publishing the work before us, Mr. Turner lays down a few geometrical definitions and illuftrations, and then proceeds to deliver the method for folving the feveral cafes in plain trigonometry by arithmetic only, in the following axioms:

"Axiom 1. Divide 4 times the square of the complement of the angle, whofe oppofite fide is either given or fought, by 300

Author of the View of the Earth;-View of the Heavens;-Syftem of Gauging

and Chronologer Perpetual.

added

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 propor tion to the hypothenuse, as that angle bears to its oppofite fide,

In angles under 45 degrees, the artificial number may be found easier thus: divide 3 times the fquare of the angle itself, whofe oppofite fide is given or fought, by 1000; the quotient added to 57.3t, a fixed number, that fum will be the artificial number required.-This is to be used, 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 root is the hypothenuje itfelf.-This it made use of, when the bafe and perpendicular are given, to find the hypothenufe.

Axiom III. The fum of the hypothenuse and one of the legs multiplied by their difference, the square root of that product will be the other leg required.-This comes into use, 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 bypothenufe, is always in proportion to 861, as the shorter leg is to its oppofite Angle. This is ufeful, when the fides are given, to find the angles.'

Why Mr. Turner fhould chufe to call the above rules by the name of axioms, we cannot imagine. An axiom implies a notion fo plain and felf-evident, that it cannot be rendered more confpicuous by demonftration: whereas the proceffes from whence fome of the above rules were deduced, are concealed; and confequently they are fo far from being axioms, that they have a fallacious appearance, and therefore require fomething more than the mere ipfe dixit of the writer to recommend them to the notice of geometricians.

But be that as it may, fuch is the method our Author has thought proper to follow, as being preferable to the logarithmical tables, which he feems to treat with contempt. We are however of opinion, that very few will follow the precepts he has laid down, and prefer a method confifting of large multiplications, divifions, and extractions of roots, to that of fimple addition and fubtraction. He tells us, indeed, that there is fome danger that the calculations by logarithms will prove erroneous, from the tables being incorrectly printed. There are doubtless fome errors in many of the logarithmical tables; but they

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

57.3 is the radius of a circle whofe circumference is 360. 86 = ruuius and half of a circle whofe circumference is 360.'

few

few, that this objection is of very little importance: and, with fubmiffion to Mr. Turner, we cannot help thinking, that there is much more danger of errors creeping into the long and tedious calculations, than in those performed by the logarithmic tables. This we think will evidently appear from the following folutions of the firft cafe by the two methods:

The acute angles and one leg given; to find the hypothenuse, and the other leg.

Given the angle at the bafe 35°. 41'; the angle at the perpendicular 54°. 19. and the base=78; to find the hypothenufe and perpendicular.

I. By Mr. Turner's Method.

(ft.) Find the Natural Radius by Axiom I.

(3d.) Find

(3d) Find the Perpendicular by Axiom III
96 To Hypothenufe

78 Add the Bafe

174 Sum multiply

18 by Difference

1392

174

Extract the Root 3132(55.9 + Perpendicular

25

1057632

525

1109)10700

998

719

Answer, {Hypothenufe, 95.9 +, or 96.

Perpendicular, 55.9+, or 56:

We fhall now leave the Reader to determine what credit fhould be given to our Author's affertion, that trigonometrical problems may be folved as expeditiously by his method as by the logarithms. But it is faid that the bulk of these tables renders them very troublesome and inconvenient; and if not at hand the operations. cannot be performed. The latter part of this objection is undoubtedly true: but is there no other method of folving trigonometrical problems? Surely there is: and we will venture to fay, that the geometrical method, or that of projection, will

always

always anfwer the artift's intention; and this requires neither tables nor tedious calculations.

It is very natural to fuppofe, when an author publishes a method tending to explode another, before well known and in general ufe, that the difcovery is his own, and founded on the firm bafis of truth. Mr. Turner, however, feems to be of a different opinion; for the method before us is neither new, nor wholly founded on geometrical principles. The process for finding an artificial radius was firft published by Mr. Henry Wilfon, in a treatise intitled Navigation New Modelled, p. 161, and which, if we mistake not, for we have not the first edition by us, appeared about the year 1715; and the process for finding the conftant number 86 (mentioned in Mr. Turner's fourth axiom) was given by Snellius, but deduced from falfe principles; the chord and tangent of the fame arch being fuppofed equal to each other: a fuppofition fo very abfurd, that the bare mention of it is a fufficient confutation. It will indeed be granted, that when the arch is fmall, the difference is inconfiderable; but then the error will augment as the arch increases, and when the latter is 40° the chord will be 68404, and the tangent 83910, the radius being 100000.

Mathematical theorems fhould always be built on the folid bafis of geometry; for otherwife, instead of conducting us along the paths of truth and certainty, they will lead us into the mazes of error and confufion.

We fhall conclude this article with obferving, that if the Reader is defirous of feeing the method given by Mr. Turner applied to navigation, he will find it in Wilfon's treatife above mentioned, and alfo in Kelly's Modern Navigator's Compleat Tutor, p. 46 to 59, fecond edit. And, with regard to the curious trigonometrical method for finding the hives of bees in large woods, mentioned in the title-page of this treatise, it was first published in the Philofophical Tranfactions, Numb. 376, by Paul Dud ley, Efquire.

Interefting Historical Events, relative to the Provinces of Bengal, and the Empire of Indoftan. With a feafonable Hint and Perfwafive to the Honourable the Court of Directors of the Eaft-India Company. As alfo the Mythology and Cofmogony, Fafts and Feftivals of the Gentoos, Followers of the Shaftah. And a Differtation on the Metempsychosis, commonly, though erroneously, called the Pythagorean Doctrine. By J. Z. Holwell, Efq; Part I. 8vo, 2s. 6d. Becket and De Hondt.