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ARTOXHA Course of Mathematics, defigned for the Life of the Officers and Cadets of the Royal Military College. By Ifaac Dalby Vol. 8vo. Price 148. Printed by W. Glendinning, for the Author.1803.

IN the elementary treatife under our prefent confideration, new principles are not to be expected; and an enquiry into the merits of the work must be confined to the selection and arrangement of the materials, and the peripicuity of the explanations. Mr. D. is a good mathematician; and we are always glad to fee a perfon of feienee write upon the elementary parts. A man who understands only arithmetic is not qualified to write a good treatife upon that fubject. The hand of a mafter is difcovered, only when the writer is able to teach higher principles. In relpect to the felection of the matter, we think that the author fhould have confined himself to thofe fubjects which are immedi ately useful to his pupils in the line of their profeffion.To be ready in the application of what they have been taugių, is the great object to be obtained; and experience fhows, that both in the navy and army, that this object is not eally accomplished, in confequence of the various other duties which they have to learn. Upon this ground, we think that Intereft, Pofition, the Method of making Logarithms, and the manner of computing the Tables of the Sines, Cofines, S in Trigonometry, might have been omitted. In a work of this kind compreffion of matter is a great object, elmar

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This volume contains Arithmetic, Geometry, Plane Trigange metry, and Menfuration; four fubjects very properly, chofen and arranged. In the preface the author obferves, that

As the Arithmetic is principally defigned for thofe who are acquainted with the firft rules, we have entered upon fractions, immediately after the divifion of whole numbers. This feems the order which naturally prefents itself, because fractions refult from the divifion of integers. The examples, therefore, in all the fubfequent branches, are indifcriminately in whole numbers and fractions 1. 76.5 ast asleds-lisw vitesil sad aviwie rodius .5

In this arrangement we agree with the author. In order to understand fractions, nothing but a knowledge of the first four rules is neceffary: there no realon, therefore, can be why the doctrine of fractions fhould not immediately follow them. The fooner they are taught the better, as but little can be done in arithmetic without them. Upon the plan of the work, Mr. D. has obferved, that

"A thorough

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"A thorough knowledge of fractions, with the proper management of the rules of proportion, will enable the ftudent very readily to comprehend nearly all that is neceffary to be acquired in arithmetic; for most of the other branches are only applica tions of the rule of three. In demonftrating or investigating by numbers, the direct rules for extracting the fquare and cube roots, we find it not eafy to avoid circumlocution, notwithstandIng the procefs of forming the powers is fomewhat algebraical. It will be perceived, that the rules in general are not fyftema tically detached from the demonstrations; this the student, whofe object is real knowledge, will not confider as a defect in method, because it may frequently prove the means of enforcing the study of principles, A more commodious arrangement might therefore have been adopted for thofe who wish to acquire the practice of arithmetic only. boops

To the propriety of thofe remarks, and of the reafons for miking the theory and practice, we give our most unqualified affent. We always with to fee theory and practice go བཀོར། hand in hand, without which we can never expect much proficiency either in fcience, or even in the practical part of it. It is very effential that the reafons of the rules fhould go along with the rules themselves; and we are glad to find that the author has followed this method. In the firft four rules the author has made their proof very clear and eafy to be comprehended, by explaining the reafons of the operation in the particular examples. We think that the multiplication table fhould, as ufual, have been extended to 12, as tending frequently to fhorten the work. In fractions, the rules are delivered with great clearnefs, and the examples are worked out at full length, accompanied with fuch explanations as tend to explain the reafons of the operations, The multiplication of Duodecimals, or as it is commonly called Crafs Multiplication, is more clearly explained than we Irave ever before feen; it is explained upon its natural principle. Tables of Money, Weights, and Measures, follow next; after which, in the natural order, comes Reduction, with Compound Multiplication and Divifion. The doctrine of Proportion is very fully confidered, and illuftrated by an excellently well-chofen fet of examples, in which, as usual, the author always keeps utility in view. The rule of Pofition, or of Falfe, as it is frequently called, is next explained, and the rule very clearly flated; for a demonftration of which, the reader is referred to the Algebra, which is to appear in a future volume. The proof could not now be made evident from the examples. In the extraction of the fquaré and cube roots, the author has given the ufual

rules, and explained the reafons from the particular in ftances; referring, however, to the Algebra for a more com plete demonftration. He has alfo added an approximating rule for the roots of pure powers, from the rational Formule of Dr. Halley. Arithmetical and Geometrical Progreffions are next treated of, and thefe the author has explained with his ufual perfpicuity. Then follows a very valuable, extenfive, and well-chosen set of examples to all the preceding rules and this part of the work (arithmetic) concludes with an explanation of the nature and ufe of Logarithms.

It is not eafy to make any confiderable abridgment of Euclid, and retain all the useful propofitions, without departing from the ftri&tnels of his principles. In the fhort fyftem of Geometry, therefore, given by the prefent author, we find fome things among the Definitions which are strictly Propofi tions, and require demonftration, and which Euclid has actually demonftrated; and if we grant them to be fuch truths as the mind readily gives its affent to upon their being propofed, we ftill think it would be better to teach Geometry upon the moft fimple and acknowledged principles, as habí tuating the mind to be more cautious in adinitting truths as felf-evident. In the doctrine of Ratios, Mr. D. feems to admit that his reafoning is not ftrictly geometrical.

"Euclid's Elements of Geometry, in the most concife form, generally make a feparate work, and are therefore too extenfive to be admitted at length into a volume of this kind. But we have endeavoured to give all the Theorems necessary for the two moft ufeful practical branches, Trigonometry and Menfuration. The latter, however, is fuppofed to include fuch figures only as depend on right lines and the circle. And with a view to facilitate the tranfition from Theory to Practice, when Ratios or Proportions are concerned, we have fometimes abridged the Demonftrations, by referring to analogous operations in Arithmetic. This may be deemed ungeometrical: but it ought to be remembered, that many who study Euclid do not wholly compre hend the doctrine of Proportion as laid down in the fifth book, without tracing the methods of Demonftration by means of an arithmetical or algebraic procefs."

In the demonftration of fome of the Propofitions, the author fuppofes lines to be drawn, without having first fhown that fuch lines can be drawn; a method which is never ufed by Euclid: if he want to bisect an angle for the demonftration of a Propofition, he has previously shown how this may be done. Though the poffibility, in this inftance, is felf-evident, yet it is not fo in all cafes. From a defi

nition of fimilar rectilinear figures, he immediately affirms all circles to be fimilar; but if the definition is to be carried from rectilinear to curvilinear figures, the application fhould furely be fhown. In the Cor. to Art, 104. it is taken for granted, that the ultimate ratio of the arc of a circle and its chord, when they are both made to vanifh, is a ratio of equality; a circumftance which undoubtedly requires demonftration; and which NEWTON has demonftrated to be true for all curves (Principia, Lib. 1. Sect. 1. Tom. 6.) In demonftrating many of the Propofitions, the arcs are made the measures of the angles; this, cannot, in strictnefs of Geometry, be taken for granted; it being a Propofition which Euclid has demonftrated, and which certainly requires proof. The converfe of Propofitions is fometimes affumed, to which we muft object. The Cor. to Art. 9. fhould have been demonftrated from the definition of parallel lines. In Geometry, whatever can be demonftrated, fhould never be affumed as a truth. The smaller number of axioms and poftulates, the better. We make thefe remarks to justify our recommendation of Euclid, as the best book for teaching Geometry. We have no objection to admit any improvements in this branch of science; but we wish never to fee the rigour of demonftration departed from. Admitting the author's first principles, we have nothing to object to his demonftrations. In the application of Geometry to "the method of tracing the figures on the ground," we find indeed much to commend: the Problems are admirably adapted to the purpofes intended. The practical methods of performing the various operations of finding the pofitions of objects, their diftances, and the lengths of lines which admit of no direct measurement; and that without the use of any inftrument for the menfuration of angles; all these things are explained with great clearness, and cannot fail to be of great ufe to thofe for whofe inftruction the work was intended. We are precluded from giving inftances from the want of figures; but the reader may be affured, that in this part of the work he will receive every afliftance and fatisfaction he can poffibly wifh. It is, perhaps, the most important practical part, and fhould be well ftudied by all whom it may concern.

The next fubject here treated of is Plane Trigonometry. Here the author, after the ufual definitions, fhows the method of computing the Sines, Cofines, &c. which, as before. obferved, we think might have been omitted, confidering for whofe ufe the work is intended. He then proceeds to explain the ufe of the tables. His proof, that" the fides

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of every plane triangle are as the fines of their oppofite angles's is lefs fimple than it might have been. Befides thefolution of the different cafes according to the data, from the principles of Trigonometry by a direct method, he has very properly fhown how it may be done geometrically, and by inftruments,dns, ade

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"Independent of computation by the tables of Sines, Tangents, &c. the feveral cafes of Trigonometry are alfo refolved geometrically and inflrumentally. A fcale of equal parts, with a line of chords, or a protractor for laying down or measuring angles, are fufficient for the geometrical conftruction, which is the moft fimple, but leaft accurate, method of folution."

The author then goes on to defcribe the Sector, giving an explanation of the various lines which are found upon it; and having premised what may be thought neceffary respecting the trigonometrical canon, and the logarithmic scale, he proceeds to refolve the feveral cafes of Plane Trigonometry. The refolution is given three different ways.. Ift. Geome trically, from a fcale of equal parts. 2d. Arithmetically, or by computation; and this is done both by the natural numbers, and by logarithms. 3d. Inftrumentally, by the logarithmic, or Gunter's fcale. In each cafe the method of refolution is explained with great clearnefs, in confequence of the arrangements of the operations. Upon the different folutions, the author makes this obfervation:

"The method of working the last proportion (when the fides are given to find the angles) by the logarithmic fcale is omitted, it being rather complex, and therefore may produce confiderable uncertainty in the refults, particularly on the fix-inch fector. We may alfo remark in general respecting these operations, that when the fides of the triangles exceed 1000, the calculations fhould be made with the pen, because there is too much guess work on the fcales when the integers are more than three.”

Having explained very fully all the principles of Trigonometry, Mr. D. proceeds to the "Application in meafur ing heights and distances."

"The inftrument proper for meafuring horizontal and vertical angles in common trigonometrical operations, is a theodolite furnished with one or two telescopes, and a vertical arc and if the horizontal is not lefs than 64 inches in diameter, the obferved angles may be read off to half a minute. Short bafes for temporary ufes only, are fometimes measured with rods, or the Gunter's chain of 66 feet. But the common 50, or 100 feet

tapes,