tinguishes the integral from the fractional part of the quantity, the value of the quantity may be augmented or diminished indefinitely in a tenfold proportion, (see page 254 of this work): but however convenient this system may be to the calculator, it is attended with very great trouble to the constructor of instruments of mensuration, and has more than any thing else retarded the progress of such instruments to perfection. For the natural mode of division, by taking successively the half of the quantity given or discovered, is susceptible of the most scrupulous exactness, whereas the division of any given space into ten equal parts can only be accomplished by repeated trials and approximations, or by complicated machines, in the original construction of which the difficulty they are intended ⚫to remove, must itself be previously surmounted.

Lines of equal parts, and of sines, tangents, secants, &c, are commonly laid down on the Gunter's Scale, so called from the name of the inventor, an eminent English mathematician, who died about 1626.

By supposing the radius of a circle to be divided into a determinate number of equal parts, the respective lengths of the sine, tangent, secant, &c. of any portion of the circumference may be ascertained, and registered in tables for the purposes of proportional calculation: but as calculations by the ordinary process of addition, subtraction, multiplication, division, and the involution or evolution of roots, are liable to become very voluminous, and consequently to be susceptible of errors of great consequence, methods have at various periods been adopted to shorten the labour and diminish the occasions of error in arithmetical operations. Of such methods, by far the most complete is the use of certain artificial numbers called Logarithms, from two Greek terms signifying the numbers of the ratios or proportions existing between other numbers with which they are connected.

If, for instance, we take a set of numbers increasing by a given geometrical progression, such as that every suc ceeding number shall be double its predecessor, as 1, 2, 4, 8, 16, 32, 64, 128, &c. and opposite to these place a set of numbers increasing by a given arithmetical progression, such that every succeeding number shall be 1 more than its predecessor, then it will be found that, by simply adding together these last numbers, the same effect will be produced as if we had multiplied the corresponding numbers of the first set; and that by subtracting the last set, the same effect is produced as if the first set had been divided, and so on with all other operations, as may be seen from the following table, where the upper row of numbers are in arithmetical progression differing by 1, and the lower row are numbers in geometrical progression, increasing in a twofold proportion.

Here the upper numbers express the numbers of the ratio of the lower row; for, if the given quantity at the begin ning of the geometrical progression be unity or 1, the first step of the progression, or 2 times 1 = 2, is expressed by the figure over the 2; the second step, 2X2=4, is expressed by 2 over 4; the third step, 2×2×2=8, is expressed by 3 over 8; the fourth step, 2x2x2x2=16, is expressed by 4 over 16, and so on to the ninth step, which is expressed by 9 over 512: consequently, the upper row of figures may be considered as the indexes of the proportions between those in the lower row, or, in other words, the

upper

tinguishes the integral from the fractional part of the quantity, the value of the quantity may be augmented or diminished indefinitely in a tenfold proportion, (see page 254 of this work); but however convenient this system may be to the calculator, it is attended with very great trouble to the constructor of instruments of mensuration, and has more than any thing else retarded the progress of such instruments to perfection. For the natural mode of division, by taking successively the half of the quantity given or discovered, is susceptible of the most scrupulous exactness, whereas the division of any given space into ten equal parts can only be accomplished by repeated trials and approximations, or by complicated machines, in the original construction of which the difficulty they are intended to remove, must itself be previously surmounted.

Lines of equal parts, and of sines, tangents, secants, &c, are commonly laid down on the Gunter's Scale, so called from the name of the inventor, an eminent English mathematician, who died about 1626.

By supposing the radius of a circle to be divided into a determinate number of equal parts, the respective lengths of the sine, tangent, secant, &c. of any portion of the circumference may be ascertained, and registered in tables for the purposes of proportional calculation: but as calculations by the ordinary process of addition, subtraction, multiplication, division, and the involution or evolution of roots, are liable to become very voluminous, and consequently to be susceptible of errors of great consequence, methods have at various periods been adopted to shorten the labour and diminish the occasions of error in arithmetical operations. Of such methods, by far the most complete is the use of certain artificial numbers called Logarithms, from two Greek terms signifying the numbers of the ratios or proportions existing between other numbers with which they are connected.

If, for instance, we take a set of numbers increasing by a given geometrical progression, such as that every suc ceeding number shall be double its predecessor, as 1, 2, 4, 8, 16, 32, 64, 128, &c. and opposite to these place a set of numbers increasing by a given arithmetical progression, such that every succeeding number shall be 1 more than its predecessor, then it will be found that, by simply adding together these last numbers, the same effect will be produced as if we had multiplied the corresponding numbers of the first set; and that by subtracting the last set, the same effect is produced as if the first set had been divided, and so on with all other operations, as may be seen from the following table, where the upper row of numbers are in arithmetical progression differing by 1, and the lower row are numbers in geometrical progression, increasing in a twofold proportion.

Here the upper numbers express the numbers of the ratio of the lower row; for, if the given quantity at the beginning of the geometrical progression be unity or 1, the first step of the progression, or 2 times 1 = 2, is expressed by the figurer over the 2; the second step, 2×2=4, is expressed by 2 over 4; the third step, 2x2x2=8, is expressed by 3 over 8; the fourth step, 2×2×2×2=16, is expressed by 4 over 16, and so on to the ninth step, which is expressed by 9 over 512: consequently, the upper row of figures may be considered as the indexes of the proportions between those in the lower row, or, in other words, the

upper

upper row contains the logarithms of the natural number's in the lower row.

Now, for example, let it be required to multiply together any two of the lower row, as 4 and 8; we find the index or logarithm of the table standing over 4 is 2, and that over 8 is 3: then, by adding together the indexes 2 and 3, we have 5, and observing the natural number placed under 5, we obtain 32, which is the product of 4 multiplied into 8.

Again, to multiply 4 by 16, and the product by 8, we add together the index of 42, the index 16-4, and the index of 8=3, in all 9, which is the index of 512, the product of 4x 16x8.

Let it be required to divide 512 by 8: the index or logarithm of 512 is 9, and that of 8 is 3; subtracting 3 from 9, the remainder is 6, which is the logarithm of 61, the quotient of 512 divided by 8, as was proposed.

To square any number, as 8, we have only to double its logarithin 3, to have 6 the logarithm of 64, the square of 8: and the cube of 8 will be found by tripling its logarithm 3, thus obtaining 9, the logarithm of 512, which is the cube of 8.

Again, to extract the square root of any number, as 64, we have only to take one half of the logarithm 6 or 3, which is the logarithm of 8, the square root of 64: and to extract the cube root of 512, we take one third part of its logarithm 9 or 3, which is the logarithm of 8, the cube root of 512.

From the consideration of such properties of a scries of arithmetical proportionals, compared with one of geometrical proportionals, John Napier, Baron of Merchiston, in Scotland, gave to the world, in 1614, a collection of tables of natural sines and sines-complement, with their corresponding logarithms, calculated for every minute of the quadrant of a circle; together with directions how, from

those