products are to be subtracted from the former, and therefore (by 5.) must be connected with them, with the signs of their terms every where changed; which proves the latter part of the corollary. The product therefore of pq+s+t into a—b+de+f is equal to the following set of products;

ap-aq+ar-as+at +dp-dq+drds+dt

+fp-fq+fr-fs+ft

-bp+bg-br+bs-bt

-ep+eg-er+es-et

¶ 15. 'Tis evident the preceding rule coincides with the common one mentioned in T3; to wit, that -X gives- —, +X—gives, +×+ gives +, and —x gives; or, that as often as the signs of the factors are alike the product must be marked with the sign +, and as often as they are unlike, the product must be marked with the sign For in multiplying pq+r-s+t by a, d, and f, which are marked with the sign+, the terms of the product have respectively the same terms of the multiplicand p-q+r—s+t; therefore, as often as the terms of the multiplicand are marked with the sign +, or the signs of the factors are alike, the products are marked with the sign +; and as often as the terms of the multiplicand are marked with the sign -, or the signs of the factors are unlike, the products are marked with the sign And in multiplying p +-+1 by band e, which are marked with the sign

the terms of the product are marked with signs. respectively contrary to those of the terms of the multiplicand ++; therefore, as often as the terms of the multiplicand are marked with the sign, or the signs of the factors are unlike, the products are marked with the sign —; and as often as the terms of the multiplicand are marked with the sign -, or the signs of the factors are alike, the products are marked with the sign +.

16. Some of these cases of multiplication are ex

pressly contained in the fifth book of the Elements, as will be evident to any one that shall examine it; which that the reader may do it with more ease, I shall here set down most of the propositions contained in it in algebraic characters, as follows. Let a and 6 represent any two quantities whatsoever of the same kind, whether commensurable or incommensurable, whereof a is the greater; p a whole number, or multiple of the quantity called unit; n a whole number, in the 2d, 3d, 4th, and 6th propositions, and in the 22d, 23d, and 24th, any quantity whatsoever, whether commensurable or incommensurable, to a unit; m a whole number, in the 1st, 2d, 3d, 4th, 5th, 6th, and 15th propositions, and in the rest, any quantity whatsoever, whether commensurable or incommensurable, to a unit;

And the first proposition will (as has been already observed in 11.) be this, ma+mb=mxa+b; The 2d will be this, ma+na: a :: mb+nb: b; The 3d this, mna: a:: mnb: b ; The 4th this, pma: na :: pmb: nb;

The 5th this, ma¬mb=mxɑ—6;

The 6th this, ma-na: a : : mb-nb : b ;

The 12th this, ma: a :: ma+mb+mc : u+btc i The 15th this, a:b:: ma: mb;

The 16th this, ma mb :: a: b;

The 17th this, ma—a : a :: mb-b: b;

The 18th this, ma+a: a :: mb+b: b;

The 19th this, ma: a:: ma-mb: a-b ;

The corollary to the 19th this, ma: ma-a:: mb:. mbb;

The 22d this, ma: na :: mb : nb ;

And the 24th this, ma+na: a :: mb+nb: b. · The demonstrations of these propositions might here be expressed in the same characters; but this, as it is not absolutely necessary, after what has been

already demonstrated in the foregoing theorem and corollaries, and may very easily be done by any one that pleases to try it, I shall leave to the industry and curiosity of the reader.

17. As it has been shown in ¶ 9. that the product of two quantities is exactly the same, whichsoever of them is made the multiplicator, I shall, in the following articles, make no kind of distinction in the notation of such products, according as one or other of the factors is considered as the multiplicator, but shall denote the product of p and q indifferently by pq or qp, whether p be multiplied by q, or q by p, as I imagine such a distinction cannot be of any use, although, for the sake of perhaps a needless accuracy in our first entrance on the subject of these articles, it has been observed in the foregoing theorem and corollaries.

18. It will be useful to add the following instances of multiplication.

a+b

03

SOLUTION.

In extracting the roots of the powers of numbers, we begin at the units place, and separate the given power into periods, each containing a number of digits, equal to the exponent of the power. This well known process being attentively considered, in conjunction with the above lemma and corollaries, it will evidently appear that the number of periods in the power is always equal to the number of digits in the root; which, from the nature of the question, was to be shown.

V. QUEST. 77. Solved by the proposer, Thomas Maughan, Quebec.

In a projection of the spherical earth, let P be the pole, and PA, PB, PC, PD, &c. meridian circles, here represented by straight lines. G Imagine a ship to sail from the point A on any oblique course PA; and she will evidently intersect the meridians at a constant angle equal

to that course. Hence her path will be a spiral line Abcdef, &c. making the angles PAb, Pbc, Pcd, Pde, &c. all equal one to another, until the vessel arrive at the pole P. The line Abcde, &c. is by the theoretical writers on navigation called the loxodromic spiral, from the two Greek words, loxos, signifying oblique, and dromos, a course.

In the annexed projection of a portion of the spherical earth, P represents the pole, and PA and PB two meridians. Let a ship from A sail on an oblique course along the loxodromic spiral Aa123, &c. until she arrive at C; through C and A draw the parallels of latitude CD and AB; and AD=CB is

D

P

the Difference of Latitude between A and C. Further, since a ship's Meridional Distance is known to be "her "distance from the meridian she has left measured "on the parallel of latitude she is in ;" it is plain that CD is the ship's meridional distance at C from A. But the Meridional Distance differs essentially from the Departure; for "Departure is the whole easting "or westing, continually made by a vessel, in steer"ing an oblique course." To illustrate this, imagine the loxodromic distance AC to be divided into infinitely small equal parts, in the points a, 1, 2, 3, 4, &c. and through these points draw the meridians Pod, Pol, Po2, Pos, &c. and ba, ol, 02, 03, &c. arcs of parallels of latitude. Then since the meridians converge towards the pole P, the Departure, ba+o1 +02+03+&c. is greater than DC the Meridional Distance. But had the ship sailed from C, on an equal and opposite course, until she arrived at A, her Departure, oC+04+03+02+&c. would have been less than AB, her Meridional Distance at A from C. Having now shown the difference between Meridional Distance and Departure, I shall next demonstrate the following general.

THEOREM.

In sailing on an oblique course, radius is to the cosine of the course, as the distance sailed is to the dif