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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 finaller space well cultivated.”

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

:: P.

Plain Trigonometry rendered easy and familiar, ly Calculations in

Arithmetic only: With its Application and Use in ascertaining all Kinds of Heights, Depihs, and Distances, in the Heavens, as, well as on the Earth and Seas; whether of Towers, Forts, Trees, Pyramids, Columıns, Wills, Ships, Hill, Clouds, Thunder and Lightning, Atmosphere, Sun, Moon, Mountains in the Moon, Shadiws of Earth and Moon, Beginning and End of Eilipjese, &c. In which is also shewr, a curious Trigonometrical Method of discovering the Places where Bees hive in large II cods, in order to obtain, more readily, the falutary Produce of thoje littie Infests. By the Rev. Mr. Turner*, late of Magdalen-hall, Oxford.

Folio. 2s. 6d. Crowder. TN a short dedication of this little treatise, the Author ob

I serves, that the common method of answering trigonometrical problems being by large tables of fines, tanyents and lecanis, renders it not only expensive by the purchase of them ;. but often precarious in the solution, by the mistakes of the press. I have therefore, adds he, for the use of the young mathematician, (from a consideration of what has been published on this curious subject) composed the present system, by which any of the cases in right or oblique plain triangles may be an." swered on the spot, by an ealy calculation in arithmetic only. The great advantages resulting from this method to gentlemen in the army or navy, as well as to those in their private studies at home, must immediately appear; as it will be found to anni swer the most necessary problems as expeditiously as logarithms; and at the same time wholly deliver you from those voluminous tables, and the inartificial fatigues of carrying them always with you.'. .

Having premised these considerations as a reason for publithing the work before us, Mr. Turner lays down a few geometri. cal definitions and illustrations, and then proceeds to deliver the method for folving the several cases in plain trigonometry by arithmetic only, in the following axioms :

Axiom 1. Divide 4 times the square of the complement of the angl o fe opposite side is either given or fought, by 300

rolle View of the Earth ;-View of the Heavens ;-System .. n Chronologer Perpetual.


added to 3 times the said complement; this quotient added to the said angle, will give you an artificial number, called sometimes the natural radius *, which will ever bear the same proportion to the hypothenuse, as that angle bears to its oppofite side.

In angles under 45 degrees, the artificial number may be found easier thus: divide 3 times the square of the angle itself, whofe opposite fide is given or sought, by 1000; the quotient added to 57.3 t, a fixed number, that sum will be the artificial mümber required. This is to be used, when the angles and a fide are given, to find another side.

Axiom IÍ. The square of both the legs, i. e. the square of the base and perpendicular added together, is equal to the square of the hypothenuse; whose root is the hypothenuje itself. This is made use of, when the base and perpendicular are given, to find the hypothenuse.

Axiom III. The sum 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 hypothenuse and one leg is given, to find the other leg.

Axiom IV. Half the longer of the two legs, added to the hye pothenuse, is always in proportion to 861, as the shorter leg is to its opposite Angle. This is useful, when the sides are given, to find the angles.'

Why Mr. Turner should chuse to call the above rules by the name of axioms, we cannot imagine. An axiom implies a notion so plain and self-evident, that it cannot be rendered more conspicuous by demonstration : whereas the processes from whence fome of the above rules were deduced, are concealed ; and consequently they are so far from being axioms, that they have a fallacious appearance, and therefore require something, more than the mere ipse dixit of the writer to recommend thein to the notice of geometricians.

But be that as it may, such is the method our Author has thought proper to follow, as being preferable to the logarithmical tables, which he seems to treat with contempt. We are however of opinion, that very few will follow the precepts he has laid down, and prefer a method consisting of large multipliCacions, divisions, and extractions of roots, to that of simple addition and fubtraction. He tells us, indeed, that there is some danger that the calculations by logarithms will prove erroneous, from the tables being incorrecily printed. There are doubtless iome errors in many of the logarithmical tables; but they are so

•• The natural radius is only turning the right angle, = go degrees, into an artificial number, which shall always bear the same proportion to the hy; othenuje, as the given angle does to its oppofite leg.

of 57.3 is the radius of a circle whose circumference is 360,
• I 86 = radius and half of a circle whose circumference is 360.'


few, that this objection is of very little importance : and, with submission 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 solutions of the first case by the two methods :

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

Given the angle at the base = 35°.41'; the angle at the per. pendicular =54o. 19 . and the base =78; to find the hypothea nuse and perpendicular.

I. By Mr. Turner's Method.
(ift.) Find the Natural Radius by Axiom L.


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{2d.) Find the Hypothenuse by Axiom I.
Angle + : Base :: Nat. Rad.
As 54.3 --78 06.8


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.....(30.) Find the Perpendicular by Axiom III. .

96 To Hypothenuse ..;. . ..78 Add the Base ".

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719 Answer,

S Hypothenuse, 95.9 +, or 96. wer, 1 Perpendicular, 55:9 +, or 56.

II. By the Logarithms.
As Radius

! 10.0000000
To the Base 78 .

- 1.8920946
So is Sec. of the Ang. 35o. 41. 10.0903085
To the Hypothenuse 96.02 - 1.9824031

As Radius

- 10.0000000
To the Base 78 ...

1.8920946 So is Tang. of the Ang. 350.41'. 9.8562042

To the Perpendicular 56.01 - 1.7482988 We shall now leave the Reader to determine what credit should be given to our Author's assertion, that trigonometrical problems may be solved as expeditiazıly by his method as by the logarithms. But it is said 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 solving trigonometrical problems: Surely there is : and we will venture to say, that the geometrical method, or that of projection, will


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on geometrica first publishe Modelled, P

always answer the artist's intention; and this' requires neither tables nor tedious calculations.''

It is very natural to suppose, when an author publishes a me-thod tending to explode another, before well known and in general use, that the discovery is his own, and founded on the firm basis of truth. Mr. Turner, however, seems 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 first published by Mr. Henry Wilson, 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 constant number 86 (mentioned in Mr. Turner's fourth axiom) was given by Snellius, but deduced from false principles; the chord and tangent of the same arch being supposed equal to each other : a suppofition so very absurd, that the bare mention of it is a sufficient confutation. It will indeed be granted, that when the arch is small, the difference is inconsiderable ; 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 should always be built on the solid basis of geometry; for otherwise, instead of conducting us along the paths of truth and certainty, they will lead us into the mazes of error and confusion. . We thall conclude this article with observing, that if the Reader is desirous of seeing the method given by Mr. Turner applied to navigation, he will find it in Wilson's 'treatise above mentioned, and also in Kelly's Modern Navigator's Compleat Tutor, p. 46 to 59, second 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 Philosophical Transactions, Numb. 376, by Paul Dudley, Esquire.


Interesting Historical Events, relative to the Provinces of Bengal,

and the Empire of Indoftan. With a seasonable Hint and Per

swalive to the Honourable the Court of Directors of the East-India * Company. As also the Mythology and Cosmogony, Fasts and Feftir vals of the Gentoos, Followers of the Shaftah. And a Dissertation

on the Metempsychosis, commonly, though erroneously, called the Py. thagorean Doctrine. By J. Z. Holwell, Esq; Part I. 8vo.

25. 6d, Becket and De Hondt. :: Rey. Od. 1765.


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