Sidebilder
PDF
ePub

to the memory of the student; and perhaps they are in reality given, but the arrangement of the present volume is so irregular, that we know not, in seeking a particular rule or formula, the parts towards which we ought to direct our search,

Napier's analogies are, however, demonstrated very neatly, after Le Gendre, in page 353; and on the plan of this demonstration, we wish that other demonstrations had been made and inserted,

In plane trigonometry, the author has furnished many for, mule that will be useful for expediting arithmetical computation, and for rapidly conducting the process of analytical dẹ, duction such as

sin. a+sin. b 2 sin, (a+b) cos. (a-b)

[blocks in formation]

To have demonstrated all these formule would perhaps have swelled the work beyond a manageable size: yet some at least ought to have been deduced from the fundamental formulæ sin (ab) sin. a cos. b±cos. a sin b;

=

and so many, that a student with moderate talents might, by his own powers and exertions, have deduced the remainder: but all the formule are not easily obtained. It is not every student who would discern the mode of deducing, for instance this form :

a

sin. (45+ ~~) = √( 1 + sin. a) (rad. = 1)

2

2

Mr. Bonnycastle has inserted the formulæ from which, in the composition of trigonometrical tables, the sines and cosines of arcs from 1° to 90° may be calculated: thus, if a and b be two arcs,

sin. (a+b) = 2 sin. a—sin. (a—b)—4 sin. a ( sin. — )*

hence let ab=1°, and we have

sin 2o2 sin. 1-4 sin. 1° (sin. 30")":

or if we put a=1′

sin. 2′-2 sin. 1'-4 sin. 1' (sin. 30")

and hence, putting a successively, 2, 3, 4, &c. we have
sin. 3°2 sin. 2°-sin. 1°4 sin. 2° (siu. 30')

sin. 4° 2 sin. 3°-sin. 2°-4 sin. 3° (sin. 30')
&c.

These sines, then, are easily formed, if the sine of 1° or of 1′′ be computed; and this computation may be effected either by

the

the continued bisection of an arc of 60°, and the extraction of the square root of quantities, or by the Newtonian series for the sine of an arc z, viz.

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

The forms that we have given are, we believe, similar to those which Prony and his coadjutors have adopted in the construction of their great trigonometrical tables; and besides these regular forms, others were employed as checks on the calculation, and as verifications: as for example,

[blocks in formation]

sin. (54°+a) + sin. (54°—a) — sin. (18o +a) — sin. (18°—a) sin. (90°-a)

&c.

Several forms, of a similar kind, are with much propriety introduced into this tract: but of these forms, with great inattention to the interests and wants of students, Mr. B. has omitted the demonstration. It is indispensably necessary that he should publish a key to his present trigonometry.

It may not be improper (says Mr. B.) to subjoin forms, &c. such as

Arc. tan. x arc. tan. arc. tan.

[blocks in formation]

x + v

I-xy

:

but we apprehend that it is improper to subjoin such, if the demonstration of them be not also supplied or if no principle, related method, or similar plan, be given, from which the demonstration may be deduced; or if not one word be said concerning the practical application, or the analytical utility of such forms. What is thus shewn, the bewildered student must hate.

Mr. B. has given, but without demonstration, the expres sions for cos. na, sin. na, in terms of sin. a, (sin. a)3 &c. ; and he then observes that the formula are equally true when ʼn is a fractional number. We hesitate to assent to this remark: it is not warranted by the principle of the processes which we have either seen used or been accustomed to use.

Of the forms, however, which express the sine and cosine of an arc by means of factors, Mr. B. has added the demonstration, and has pointed out the application and practical utility. They are

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

Of these formulæ, we believe (for we possess not the means of immediate reference) that Euler was the author; Waring has noticed them; and they are inserted in the Introduction to Callet's Table of Logarithms. The exponential expressions for sin. x, &c. are also employed in demonstration: but what it is that determines the present author to his occasional fits of demonstration we are wholly unable to decide: it is neither the pre-eminent difficulty nor the importance of the thing demonstrated.

With respect to the astronomical part, we may say, as we hate already said of the trigonometrical, that it contains many useful rules and appropriate examples, but not the due quantum of demonstration. In the problem by which the moon's distance is cleared from the effects of parallax and refraction, the rule is given, but the demonstration is suppressed; yet it might have been comprised nearly within the space preceding the rule, in which the author explains by what mode of solution, not the mode on which the rule is founded, the true distance may be investigated.

At the end of the astronomical part, Mr. B. has added some astronomical problems to be solved by the student: but such only ought to have been given as come within the scope of preceding rules, or within the application of preceding principles. Where is the doctrine or theorem in the present voJume, that enables us to find the latitude of the place at which a degree on the meridian is equal to a degree on the equator, the earth being a spheroid of given dimensions?

In the Introduction, the author has neatly and concisely stated the rise and origin of the logarithmic and trigonometrical calculus, and has attributed to the respective authors the honours of their discoveries. After having taken notice of the several trigonometrical tables that have been published, he adds:

To this brief account of the works of some of the early writers on this subject, and the tables which, at different times, have been composed for facilitating its practical operations, it may also be proper to subjoin a slight sketch of the improvements which it has undergone in passing through the hands of the later analysts, who, by means of a more commodious algorithm, and the resources of a ready and comprehensive calculus, unknown to their predecessors, have enlarged the boundaries of the science, and simplified its rules and pro

cesses,

• These

[ocr errors]

These advantages, and the consequent discoveries which attended them, have chiefly arisen from the new views of the subject that had been opened to mathematicians by the theorems first given by Vieta, for the chords of the sums, differences, and multiples of arcs and their supplements; which though left without demonstration, and, in the latter case, probably formed by induction from the law of the terms and their co-efficients, have, nevertheless, been the germ of most of the numerous and elegant formula which have since enriched this branch of the subject.

We are also, in this respect, no less indebted to Napier, not only for his admirable discovery of logarithms, but for the new and excellent analogies which he introduced into that part of the science relating to the solution of spherical triangles, which still go by his name; as likewise for his other well-known rules, called the Five Circular Parts; which, though too artificial and restricted to be generally employed in the present advanced state of the science, are sufficient proofs of the skill and address with which he investigated every branch of a subject so intimately connected with the invention that has gained him such just celebrity.'

The meaning of some of these latter sentences we do not rightly apprehend.Too artificial and restricted to be generally employed in the present advanced state of the science?" if the rules and formula be restricted, they cannot be generally applied but they are useful and commodious, we conceive, because they are artificial.-The succeeding passages are judiciously drawn up:

The works of Briggs, Arithmet. Log. and Trig. Britann. also greatly contributed to the advancement of this branch of the science, both by the assistance which they afforded to the practical calculator, in many intricate and difficult computations, and by the numerous improvements and discoveries of a higher kind, with which they abound. The method, in particular, which he appears to have first used in raising logarithms from their differences, and his skilful application of analytical principles to several subjects of difficult investigation, entitle him to rank with the first mathematicians of the time in which he lived.

The logarithmic and other curves, likewise, which first began to be introduced about this time, greatly facilitated the conception of those numbers, by exhibiting some of their most remarkable properties in a more perspicuous way than could be done by the abstract methods of investigation employed by Napier and others. And though the doctrine itself has no necessary connection with these or any other geometrical figures, it was from this source that the new and advantageous mode of expressing logarithms by series was first derived.

This happy improvement, which was introduced into the science about the year 1668, by Mercator and James Gregory, who were led to the discovery of some of the most simple forms of these series by contemplating the nature of the hyperbola, was soon afterwards

extended

extended to the trigonometrical part of the subject, or the arithmetic of sines, which Newton, Leibnitz, the Bernoullis, and others, enriched with similar formule; and by this means assimilated the principles of logarithms and trigonometry with those of the new calculi, of which they were the inventors and improvers.

The exponential formula, also, for the sines and cosines of arcs, which were first given by Demoivre, have greatly contributed to the progress of the analytical branch of this subject, by abridging its operations, and shortening the labour of investigation; and though some writers have represented expressions of this kind as founded upon principles which are repugnant to all our ideas of magnitude or quantity, yet their commodious form, and the ease and certainty with which they can be applied in many intricate inquiries, will always cause them to be regarded by the skilful analyst as an important acquisition to the science.

• Many other improvements, of more or less importance, have since been made, both in the practical and theoretical branches of this subject, by later writers; but of these, none have proved of such general advantage to the science as the substitution of the analytical mode of notation in place of the geometrical; which useful change was first introduced by Euler; who, besides this simplification of the former methods, has developed and extended, in his numerous works, almost every part of the trigonometrical analysis; which, under his masterly hand, assumed the form of a new science.'

Mr. Bonnycastle, perhaps, will think that we have indulged in too great strictness of criticism; and, as he may not like our censures, he will not, probably adopt our suggestions or profit by our admonitions: yet, in the present instance, hints and advice ought not to be neglected, since, if properly given and properly followed, they may be efficacious in transmuting the present work into the most useful trigonometrical treatise in our language. Our advice, hints, &c. may all be comprized in one word, demonstrate :-Fill up the skeleton of dry formulæ and rules with the flesh, blood, and marrow of demonstration. From the specimens afforded us, we presume that the author is fully competent to this task; and though it may be a task, it is necessary to be performed, in order that this production, instead of being only a register of methods to aid the memory of the experienced mathematician, or a collection of rules for the use of the merely practical operator, may aspire either to more extensive or to more real utility, and may instruct students.

ART. IX. Essays on various Subjects. By J. Bigland. 2 Vols. 8vo. 128. Boards. Longman and Co.

ON

N former occasions, we have stated that the composition of Essays is a species of writing, which in our judgment ought to be attempted only by men of strong powers of think

« ForrigeFortsett »