volume of Hodgeson's System of Mathematics, printed in the year 1723, wherein he will find that a right-angled or quadrantal spherical triangle may be stereographically constructed, not only one way, but one hundred and twenty different ways ! “How far Mr. Bonnycastle may judge it right” (says the reviewer) “ to enter into any altercation with Mr. Keith, in consequence of the liberties he has taken with his character, is not for me to decide.” If what I have charged Mr. Bonnycastle in the Philosophical Magazine) with copying from my work, has really not been copied, and can be found in any other treatise, prior to the first publication of mine, then, indeed, I have accused him unjustly, and shall, undoubtedly, meet with that contempt, which so base a conduct would merit. But if, on the contrary, my charges are well founded, and the matter in question cannot be found in any other publication (however insignificant or unimportant the subjects which are copied may be); I stand upon a foundation not to be shaken by the false assertions and illiberal insinuations of this, or of any other critic. The reviewer concludes his letter, by paying a handsome compliment to his own superior talents and acquirements, with which he seems to be highly pleased; and, as it is my fixed determination never to notice him again, I shall take my leave of him whilst he remains in that happy state of mind. With regard to Mr. Bonnycastle, who now holds one of the first mathematical situations in the kingdom, if he can conscientiously acquit his conduct of reprehension, I envy not the happiness of his disposition, but leave him in quiet possession of the honour and emolument which he may have acquired. THOMAS KEITH. ADVERTISEMENT TO THE FOURTH EDITION. THIS is edition of the ensuing work has been carefully revised. All the astronomical examples which depend on the Nautical Almanac, are adapted to the year 1822, and they may be solved either with or without the Almanac; because the several articles necessary to be taken from that work are given at the end of the different examples. By these means the students will readily learn the use of the Nautical Almanac, and in a school where several are studying the same subject, their progress will not be retarded by waiting for the Almanac. The examples will be found sufficiently numerous and appropriate for the purpose of instruction. They might easily have been extended, with different dates, by a selection from various authors; but such examples are perfectly useless to a student, who has not in his possession a Nautical Almanac corresponding with each date. No. 1. York-buildings, New-road, St. Mary-le-bone, London, October, 1820. THE CONTENTS. THE NATURE AND PROPERTIES OF LOGARITHMS. CHAP. II. THE USE OF THE TABLE OF LOGARITHMS 1. To find the logarithm of any whole number, or mixed decimal, consisting of one, two, three, or four figures 4 2. To find the logarithm of any whole number, or mixed decimal, to five or six places of figures 3. To find the logarithm of a pure decimal 4. To find the logarithm of a vulgar fraction 5. To find the number answering to any logarithm to 6. To find the number answering to any logarithm, to 7. To find the product of two whole or mixed numbers 7 8. To divide one number by another 9. To involve a number to any power; that is, to square, 10. To extract the square or cube root, &c. of any number 10 11. To find the value of a quantity having a vulgar frac- 12. To find a fourth proportional to three given num- 1. To find the natural sine or cosine of an arc, also the logarithmical sine, tangent, secant, &c. 2. To find the logarithmical sine, cosine, &c. of an arc 3. To find the degrees, minutes; or degrees, minutes, and seconds, corresponding to any given logarith- 4. To find the natural or logarithmical versed sine of |