WHEN we consider the great exertions of learned men to disseminate mathematical information in other countries, we must be surprised to find that this kind of knowledge is most shamefully neglected in the United States of America. The mathematical sciences are the foundation of almost every art that is necessary to promote the comfort and convenience of civilized man: their extensive use in human affairs stands attested by the wise and learned of every age. It cannot therefore be denied that these sciences ought, for the sake of their usefulness, to be diligently studied and liberally encouraged in a country like this. But these sciences are as valua ble in their own nature as they are useful. For by mathematical exercise, as the celebrated Dr. Barrow observes, the mind is inured to a constant diligence in study, delivered from a credulous simplicity, strongly fortified against the vanity of scepticism, restrained from rash presumption, inclined to a due assent, subject to the government of right reason, and inspired with resolution to combat the unjust tyranny of false prejudices. In Europe small periodical publications have contributed much to the diffusion of this kind of learning, and many of the greatest scientific characters of the present age have at an early stage of life commenced their mathematical career by answering the questions proposed in such works. Ambitious of seeing their little productions in print, they studied with an ardour which nothing else could inspire; and this ambition properly encouraged, by a judicious parent or preceptor, accelerated the improvement of the pupil and formed the basis of future eminence. The learned Dr. Hutton in a late publication has declared that

a small periodical work, entitled, the "Ladies' Diary,” had produced more mathematicians in England than had been done by all the mathematical authors in that kingdom. But this is not the only advantage that has resulted from such works. They have served as useful vehicles in conveying ideas, discoveries, and improvements, from one mathematician to another, and to the community at large. By these means the limits of science have been extended, and the common stock of mathematical knowledge continually augmented and enriched. The Mathematical Correspondent will be conducted on the same plan as the European works to which we have just alluded; and as similar causes generally produce similar effects, we are not without hopes of rendering some service to the public.

A number of this work containing one sheet of paper, will be regularly published four times a year, viz. on the first days of May, August, November, and February. In each number a prize question will be proposed, and whoever gives the best solution to that question, one month previous to the publication of the next succeeding number, shall receive a handsome silver medal on which is the following inscription, "From the Editors of the Mathematical Correspondent, to A- B- " as a reward for his mathematical merit." The small profits arising from the sale of the work will be applied to defraying the expence of the prize medals.

With a view of rendering the work as useful as possible to the generality of our subscribers, we have begun with the lowest parts of the mathematics; and for this reason the questions proposed in the first number are such as may be resolved by a very small degree of Mathematical knowledge. In future, however, we shall gradually ascend towards the higher regions of those sciences, as far as may be thought consistent with the abilities of our readers. On this subject the opinion and advice of every American mathematician, will always be thankfully received and duly considered.

MATHEMATICAL CORRESPONDENT,

&c.

ARTICLE I.

A New elucidation of the principles of the Rule of Proportion, in Arithmetic ; applied to the resolution of practical questions, and to the invention of general rules for making in the neatest and shortest manner possible, many of the most useful calculations that daily occur in the counting-house. By G..BARON.

1. Any number may be represented by as many dots placed in a straight line

as that number contains units. A Thus the number 7 is repre

sented by seven dots placed in the annexed straight line A,

§ 2. A greater number is a multiple of a less, when the greater contains the less a certain number of times exactly. Thus

the number 12 repre- B sented by B is a multi- C ple of 4 represented by

C; for 12 contains 4 a certain number of times exactly (viz. 3 times). For the same reason 14 is a multiple of 7, 18 is a multiple of 6, &c. &c.

§ 3. When a greater number is a multiple of a less, the less is said to be a part of the greater. Thus the number 4 is a part of (its multiple) 12, 7 is a part of 14, 6 is a part of 18, &c. &c.

§ 4. The denominator of a part of any number, is the number which expresses how often that number

A

contains the part. It is called the denominator, because the name or denomination of the part is always derived from it. Thus 4 is named the third part of 12 from the denominator 3, which expresses how often 12 contains its part 4, 2 is denominated the seventh part of 14 from the denominator 7, which expresses how often 14 contains its part 2, &c. &c.

A

B

§ 5. When a number B is contained in one of its múltiples A, as often as another number D is contained in one of its multiples C; then C is said to be the like multiple of D that A is of B; and D the like part of C that

C

D

Bis of A. Thus 9 is a multiple of 3, and 12 is a like multiple of 4; 3 is a part of 9, and 4 is a like part of 12. Again 18 is a multiple of 2, and 27 is a like multiple of 3; 2 is a part of 18, and 3 is a like part of 27, &c. &c.

§ 6. When four numbers have the properties expressed in any one of the three following cases, they are proportional; and when four numbers are proportional they have the properties contained in some one of the same cases. *

CASE I.

When the first number is a multiple of the second, and the third is a like multiple of the fourth.

ILLUSTRATION.

The four numbers 6, 2, 12, and 4, are proportional, for the first is a multiple of the second, and the third is a like multiple of the fourth. Four proportional numbers are universally written thus, 6: 2 :: 12 : 4;

*This fection must be confidered, not as a definition, but merely as a defcription of the three properties, by which proportional numbers are common known. These three properties are actually three different propositions de duced from the true definition of proportional numbers. This definition is referved as a curious fubject for a future enquiry.

and read thus, 6 is to 2 as 12 is to 4. For the same reason 9: 3 :: 15: 5, 22: 2 :: 33: 3, &c. &c."

CASE II.

When the first number is a part of the second, and the third is a like part of the fourth.

ILLUSTRATION.

The four numbers 2, 8, 4, and 16, are proportional; for the first is a part of the second, and the third is a like part of the fourth. For the same reason, 3: 15 :: 5:25, 11:44 9: 36, &c. &c.

CASE III.

When the first number is a multiple of a part of the second, and the third is a like multiple of a like part

of the fourth.

The four numbers, 6, 8, 9, and 12, represented by A, B, C, and D are proportional. E is a part of the

For

ILLUSTRATION.
A..|..|

B

C

second B, and F is a like part of the fourth D; and A is a multiple of (E) a part of B, and C is a like multiple of (F) a like part of D. Consequently A: B :: C: D.

In the same manner it may be shewn that 9: 15 :: 12: 20, that 21: 14 :: 36: 24, that &c. &c.

When A, B, C, and D, represent any four proportional numbers, the first and fourth, A and D, are called the extremes, and the second and third, B and C, the means. The last number, D, is said to be a fourth. proportional to A, B, and C; and the numbers A, B, C, and D, are denominated the proportional terms.

COROLLARY.

When A: B:: C÷D, according to case III. of § 6 ;