greater part exceeds the lesser by 9, it must be equal to 22. pp. 13, 14.

The 2nd, 3rd, and 4th lectures contain the application of the four fundamental rules of arithmetic to

Algebraic quantities both in their integral and fractional form, together with the raising of powers and extraction of roots; the whole being illustrated by a variety of plain and easy examples. The next three lectures are upon simple and quadratic equations; and to assist the learner in the solution of the questions and problems contained in them, we are "that great informed, Pref. p. 3. care has been taken, that those questions whose solutions are given at full length, should bear a near resemblance and analogy to such as are left unsolved at the end of each section."

The 8th, 9th, and 10th lectures are upon the doctrine of ratios, proportion and variable quantities; the 11th and 12th upon Arithmetical and Geometrical progression; the 13th, on the arithmetic of surd quantities; these lectures, like the former, abound with a variety of easy examples and familiar questions to exercise the learner upon the several subjects contained in them.

Logarithms form the subject of the 14th lecture, upon which Mr. B. observes, Pref. p. 5. that, " in an elementary treatise of this kind, he has not thought it necessary to trouble the learner with rules for the construction of Logarithmic tables; but that his object is to explain the nature and properties of the logarithms themselves as they stand in those tables, and to apply them to several very useful arithmetical purposes." The principal purposes to which Mr. B. has applied them, are, the summation of Geometric series, the calculation of

miscellaneous subjects; such as Unlimited problems; the properties of numbers; Permutations and combinations; the different modes of in

vestigating the Binomial Theorem; the solution of Exponential equations; &c. The following is Mr. B's manner of considering the expression for a Perfect Number.

The properties of numbers which we have just now exhibited have been deduced from the manner in which any number may be algebraically represented, according to the value of its digits in the common arithmetical scale. But there are certain numbers, which may be considered as arising from the continued multiplication of other numbers. The most general form under which numbers of this kind may be presented, is an bmcr ds &c. where a, b, c, d, &c. are prime numbers, and n, m, r, s, &c. any whole numbers whatever. The divisors of an bm cr ds, &c. are 1, a, a2, a3..........an; b, ab, a2b, a3b ....an b; b2, ab2, a2b2, a3b2....an b2; &c.; c, ac, a2c, a3c....an c; &c.; ab,

abc, abc2, abc3, aber ; &c. &c. &c. &c.;

and the number of these divisors will depend upon the number af factors a, b, c, d, &c. and the magnitude of the indices n, m, r, s, &c. The most simple form under which an bm er ds &c. can be exhibited, will evidently be, when there are only two factors, and when the index of one of them is equal to unity: this form is an b, and its divisors are 1, a, a2, a3, a1....an; and b, ab, a2b, a3b,....an-1b.

Considering the composition of a number under the simple form anb, we are furnished with a method for investigating the expression for what is called a perfect number, i. e. a number which is equal to the sum of all its divisors. For in this case anb=1+ a +a2+a3....+a"; +b+ab+ a2b+a3b....+a"-1b= (by Sect. a+1-1 ab-b + a-l

73.)

a-1

.. a+b

—a"b=a+1-1+a"b—b, or

by the supposition, b is a prime number; ..a"+'-2a+1 must be equal to unity; and consequently at-2a"=0; .. a-2=0, or a=2; and in this case ba+1-1=2+1. 1; hence the expression a"b

--

becomes 2" x 2"+1-1 where 2"+1-1 must be a prime number.

Let n=1, 2, 3, 4, 5, 6, &c. then 2"+1=3, 7, 15, 31, 63, 127, 255, &c.; of which the prime numbers are 3, 7, 31, 127, &c. and the corresponding values of n are 1, 2, 4, 6, &c.; hence the perfect numbers are generated in the following order;

admitted to be just than allowed to be pleasing. His superfluous minuteness of description is successfully imitated, but the lover of the ridiculous will be disappointed if he expect to find much to make sport of in the poetry of Mr. Crabbe, who is indeed rather a moral writer in verse than a poet.

Three dramatico-medical cases with the signature of Momus Medlar.

We must pronounce the mental health of that patient to be desperate, to whom either of these cases can apply. Though little skilled in these affections of the brain, we do not hesitate to declare the most alarming symptom to be a plethora of Folly, which we shall be most happy to relieve by the use of our critical lancet. The application of ludicrous terms, and the introduction of familiar images to incidents in themselves serious and affecting, will always make the vulgar laugh, but readers of sensibility love not to be robbed of "the Joy of Grief."

We cannot think either the penitence of a deluded wife, the forgiveness of a christian husband, the horrors of a guilty usurper, or the catastrophe of a murderer, proper subjects to form the basis of a comic entertainment.

GYMNASIUM sive SYMBOLA CRI

TICA. By the Rev. ALEXANDER
CROMBIE, L. L. D. 2 vols. 8vo.
London, Johnson. 1812. Price 18s.

IT is observed by the author of the work before us, that the most successful means of becoming critically acquainted with any foreign language, is to employ it, either in composition, or translation, under the direction of some skilful teacher. A capacity of translating our vernacular tongue into another language, with accuracy and elegance, affords, he observes, the most incontestible evidence of a perfect acquaintance with the grammar, the idiom, and the purest phraseology of that language.

To assist the classical student in acquiring a correct Latin prose style, is the sole, but not unimportant purpose of the " Gymnasium sive Symbola Critica." The work, accordingly, consists of critical observations of a miscellaneous nature, but chiefly philological, illustrated by Exercises, progressively adapted to the capacity of the scholar. To enable our readers to form a judgment of these volumes, we shall first present them with a sketch of the author's plan, and then select a specimen of its execution.

The work is introduced with some preliminary observations, the purpose of which is to furnish the reader with some general rules for These writing correct Latin prose. occupy 107 pages.

The several excellencies of style, the author observes, result from a combination of the three following requisites-1st. A judicious selection of words; 2d. A natural and lucid arrangement; and 3d. An observance of those grammatical relations among the words themselves, which reputable and general

usage has established. Presuming that the reader has already acquired a competent knowledge of Latin syntax, he proceeds, in Chap. 1. of the Preliminary Observations, to offer a few directions for the selection of words. Here he is naturally led to explain the character of Barbarism; and endeavours to establish certain rules, by which the junior reader may determine the relative value of those synonimous words, which the Lexicographer may, in any case, present to his choice. After showing that Barbarisms, though sometimes admissible, should generally be avoided, as unfriendly to perspicuity, he proceeds to caution the scholar against the employment of any Latin word in a barbarous sense,also against the use of any phraseology purely poetical, and likewise against the employment of equivocal words whenever ambiguity of meaning is to be apprehended. These admonitions are accompanied with examples.

In Chap. II. which is divided into two sections, the author treats of arrangement, or the collocation of words in clauses and sentences. The first section is devoted to the consideration of comparative arrangement. Here the author, after illustrating the superiority, which a trans positive language possesses over one, which is purely analogous, in respect to arrangement, proceeds to exemplify the difference between the English and the Latin collocation, and to inquire, which of them should be deemed the natural order. Section 2. treats especially of Latin arrangement, and contains a variety of rules for the direction of the scholar in the collocation of words.

In Chap. III. the author treats of translation, and inquires, what are the qualifications necessary to

constitute the excellence of a good translator. translator. He then proceeds, in Chap. IV. to explain the causes which render translation, in many instances, imperfect, and in others, impracticable. Chap. v. is given wholly to the subject of AngloLatin translation. After observing, that idiomatical expressions should be studiously avoided, unless the idioms of the two languages should accidentally concur, and that phrases should be rendered by phrases, none being allowable, unless sanctioned by positive authority, he observes, that tropes and figures cannot always admit a close or literal translation.

In translating, it is necessary to observe, that tropes and figures cannot always be transferred from one language into another; in other words, the figurative terms cannot, in every instance, be literally translated. For example, the Latins said, Ferro occisus est, to denote, "He was slain by a sword;" but we cannot transfer the synecdoche, and say in English," he was slain by iron." To explain the origin of tropes and figures, as partly created by necessity, partly adopted from convenience, and partly introduced as subservient to beauty, elegance, vivacity, and strength, would lead us too far from our present purpose. Suffice it to observe, that tropes and figures being founded in the relation, which one object of thought bears to another, and the laws of mental association being the same in all men, a very close resemblance exists between different languages, in respect to the figurative employment of words. Hence a term, figuratively used in one language, frequently admits a literal translation into another, without violating the figure. Of this fact, it would be easy to adduce a great variety of examples. We shall content ourselves, however, with the few following

We say, in English, "The pillar of the family," denoting by metaphor,