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A short Explanation of the TABLES of LOGARITHMS,

and Sines and TANGENTS.

S

UPPOSING now the young geometrician acquainted with the Elements of Euclid, that he may

like. wife be acquainted with the Practice, as well as the Elements of Trigonometry, for which the Tables of Logarithms of Numbers, of Sines, Tangents, &c. are neceffary; but, not being fo far advanced in mathematics as to understand the proper method of constructing these tables, that depending in a great measure on infinite series ; although, at the fame time, as much is contained in the Elements of Trigonometry as is fufficient to explain the nature of these, I shall

only here give a short definition of Logarithms, and show their different uses in the practice of Trigonometry, &c.

Logarithms, the invention of Lord Napier, published in the year 1614, improved by himself and Mr Brigs of Oxford College, are such an arrangement of artificial numbers, with natu. ral ones, that the addition of the artificial numbers answer to the multiplication of the natural ones, in such a series as is judged most convenient by the calculator. That series which at present is judged most commodious is that which increases in a tenfold proportion, as I; 10, 100, 1000, 10000, &c. the artificial numbers answering to these, are, 0, 1, 2, 3, 4, &c. which are a series of numbers, in arithmetical progression, beginning with o, as above, which are the exponents or the indices of the former ; the addition or subtraction of which indices answer to the multiplication or division of the numbers, as 043+1+3=2+2, &c. which are likewise the indices of IX10000=10X1000=100 X100, &c. any riumber of arithmetical means taken betwixt I and 10, 10 and 100, &c. answering to the intermediate nacural numbers, in such manner as the addition of the one answers to the multiplication of the other, taken to any number of places, constitute a Table of Logarithms, which, taken instead of their correspondent numbers, many tedious operations are evited; e. g. if two numbers are to be multiplied together, add their log. ; if to be divided, subtract the log. of the divisor from the log of the dividend ; if a number is to be squared, double its log. ; if to be cubed, multiply its log. by 3; if the square-root is to be extracted, half its log, if the cube root of any number is to be extracted, take of its log. the number answering to which is the root required; the fame of any other power. In finding the log, answering to any number, if the number confift of one place, the exponent is o; if of 2 places, is I ; if of 3 places, 2, &c.; and, if the log. of any number have its

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A short Explanation of the TABLES of LOGARITHMS,

and Sines and TANGENTS.

S

UPPOSING now the young geometrician acquainted with the Elements of Euclid, that he may

like. wise be acquainted with the Practice, as well as the Elements of Trigonometry, for which the Tables of Logarithms of Numbers, of Sines, Tangents, &c. are neceffary ; but, not being so far advanced in mathematics as to understand the proper method of constructing these tables, that depending in a great measure on infinite series ; although, at the fame time, as much is contained in the Elements of Trigonometry as is sufficient to explain the nature of these, I shall only here give a short definition of Logarithms, and show their different uses in the practice of Trigonometry, &c.

Logarithms, the invention of Lord Napier, published in the year 1614, improved by himself and Mr Brigs of Oxford College, are such an arrangement of artificial numbers, with natural ones, that the addition of the artificial numbers answer to the multiplication of the natural ones, in such a series as is judged most convenient by the calculator. That series which at present is judged most commodiouş is that which increases in a tenfold proportion, as 1; 10, 100, 1000, 10000, &c. the artificial numbers answering to these, are, 0, 1, 2, 3, 4, &c. which are a series of numbers, in arithmetical progression, beginning with o, as above, which are the exponents or the indices of the former; the addition or subtraction of which indices answer to the multiplication or division of the numbers, as 04=+1+3=2+2, &c. which are likewise the indices of IX10000=10X1000=100 X100, &c. any riumber of arithmetical means taken betwixt I and 10, 10 and 100, &c. answering to the intermediate nacural numbers, in such manner as the addition of the one answers to the multiplication of the other, taken to any number of places, constitute a Table of Logarithms, which, taken instead of their correspondent numbers, many tedious operations are evited; e. g. if two numbers are to be multiplied together, add their log. , if to be divided, subtract the log. of the divisor from the log. of the dividend ; if a number is to be squared, double its log. ; if to be cubed, multiply its log. by 3; if the square-root is to be extracted, half its log. , if the cube-root of any number is to be extracted, take of its log. the number answering to which is the root required; the fame of any other power. In finding the log, answering to any number, if the number confist of one place, the exponent is o; if of 2 places, is 1 ; if of 3 places, 2, &c.; and, if the log. of any number have its

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