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A fhort EXPLANATION of the TABLES OF LOGARITHMS, and SINES and TANGENTS.

UPPOSING now the young geometrician ac

S quainted 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 conftructing thefe tables, that depending in a great measure on infinite feries; although, at the fame time, as much is contained in the Elements of Trigonometry as is fufficient to explain the nature of thefe, I fhall only here give a short definition of Lo garithms, and fhow their different ufes 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 fuch 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 fuch a feries as is judged most convenient by the calculator. That series which at prefent is judged most commodious 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 progreffion, beginning with o, as above, which are the exponents or the indices of the former; the addition or fubtraction of which indices answer to the multiplication or divifion of the numbers, as 04 +1+3=2+2, &c. which are likewife the indices of 1×10000=10X1000=100 X100, &c. any number of arithmetical means taken betwixt I and 10, 10 and 100, &c. answering to the intermediate natural numbers, in fuch manner as the addition of the one anfwers to the multiplication of the other, taken to any number of places, conftitute a Table of Logarithms, which, taken instead of their correfpondent numbers, many tedious operations are evited; e. g. if two numbers are to be multiplied together, add their log.; if to be divided, fubtract the log. of the divifor from the log. of the dividend; if a number is to be fquared, double its log.; if to be cubed, multiply its log. by 3; if the fquare-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, anfwering to any number, if the number confift 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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A fhort EXPLANATION of the TABLES of LOGARITĖMS, and SINES and TANGENTS.

UPPOSING now the young geometrician acquainted with the Elements of Euclid, that he may likewife 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 conftructing these tables, that depending in a great measure on infinite feries; although, at the fame time, as much is contained in the Elements of Trigonometry as is fufficient to explain the nature of thefe, I fhall only here give a short definition of Logarithms, and show their different ufes 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 fuch 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 fuch a feries as is judged moft convenient by the calculator. That feries which at prefent is judged most commodious is that which increases in a tenfold proportion, as 1; 10, 100, 1000, 10000, &c. the artificial numbers anfwering to thefe, are, o, 1, 2, 3, 4, &c. which are a series of numbers, in arithmetical progreffion, beginning with o, as above, which are the exponents or the indices of the former; the addition or fubtraction of which indices answer to the multiplication or divifion of the numbers, as 04=+1+3=2+2, &c. which are likewife the indices of 1X10000=10X1000=100 X100, &c. any number of arithmetical means taken betwixt I and 10, 10 and 100, &c. answering to the intermediate natural numbers, in fuch manner as the addition of the one anfwers to the multiplication of the other, taken to any number of places, conftitute a Table of Logarithms, which, taken instead of their correfpondent numbers, many tedious operations are evited; e. g. if two numbers are to be multiplied together, add their log.; if to be divided, fubtract the log. of the divifor from the log. of the dividend; if a number is to be fquared, double its log.; if to be cubed, multiply its log. by 3; if the fquare-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 anfwering to which is the root required; the fame of any other power. In finding the log, anfwering to any number, if the number confift 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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