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crossing the space next the upper line, it must have passed through one tenth part of that small division; in passing over the second parallel space, it must have advanced two tenths of the small division; in passing over the third parallel space, it must have advanced three tenths of the small division, and so on, until having crossed the whole ten parallel spaces, it must have advanced the whole of the small division on the upper line.

Were it now required to take in the compasses one inch, four twelfth parts, and three tenths of a twelfth, we would place one foot of the compasses on the upper line at the great division marked 1, and opening up the other towards the right hand to the point marked O, we would have a distance of one inch: then, to obtain the four twelfth parts, the same foot would be opened to the right hand until it extended to the small division marked 4; but to obtain the three tenths of one twelfth part, we move the compasses down across three parallel spaces, and stopping one foot at the line marked 3 at the right end of the scale, where it is cut by the division from one inch, and extending the other towards the diagonal proceeding from the fourth small division, we shall find the opening of the compasses must be a little increased to touch the intersection of that diagonal with the third parallel; when the space contained between the feet of the compasses will be one inch four twelfth parts, and three tenths of one twelfth part of an inch, as was required.

In this example the divisions of the scale have been considered according to their real value, the great divisions being in fact inches, and the small divisions lines, or twelfth parts of an inch; but as in laying down figures on paper such real measures can very seldom be employed, such as are strictly proportional must be adopted; for instance, if instead of considering the space between 0 and the left hand division 1 to be one inch, we suppose it to represent

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one foot, then each of the small divisions on the right of O must be considered as twelfth parts of a foot, that is, as inches, and the distances indicated by the advancing of the diagonal lines across the ten parallel lines will corre spond to tenth parts of an inch; so that the opening of the compasses, obtained in the former example, would now represent one foot four inches, and three tenths of an inch, Again, if each of the small divisions on the upper line of the scale from 0 to the right as far as 12, were supposed to represent one foot, then the great division for Q to the left as far as 1, would indicate twelve feet, and each of the advances made by the diagonals across the ten parallels would correspond to one tenth of a foot, in which case the opening of the compasses obtained in the first example would represent 12 and 4, or sixteen feet and three tenths of a foot.

The scale beginning on the upper line AA being divided into twelve equal parts, is purposely adapted for measures of feet, inches, &c.; but the scale beginning on the lower line BB is divided in a different proportion. The great divisions, numbered 1, 2, 3, 4, 5, 6, from left to right, are in fact each one half of an inch, but this real value is not taken into consideration. The small divisions at the left end are tenth parts, from each of which diagonal lines are drawn upwards to the opposite side of the scale, in the same manner as was done in dividing the great division at the right end; for the first diagonal commences at O on the lower line, and is drawn to the first division on the upper line; the second diagonal commences at the first division on the lower line, and is drawn to the second in the upper line, and so on to the end of the divisions.

Now, if we suppose one of the great divisions on the lower line BB to represent an unit of any sort of measure, as an inch, a foot, a yard, a mile, a league, &c. each of the small divisions will represent one tenth part of that unit,

and

and the advances made by the diagonals, across each of the ten parallels, will be one tenth of one tenth, that is, one hundredth part of the unit or great division. Again, let one of the great divisions represent 100, each of the small divisions must represent the tenth part of 100, or 10, and each step of the diagonals across the parallels will be the tenth of a tenth, or the hundredth part of the great division, which is 1 unit. Let it now be required to take off from this lower scale a quantity corresponding to 365 feet. We place one foot of the compasses at the great division marked 3 on the line BB, and opening up the other foot to the left until it come to O, we have 300 feet; next extending the foot still farther to the left to the sixth small division on the same bottom line, we obtain 360 feet; then moving the compasses upwards along the third great division to the fifth parallel, and opening them a little more, so as to touch the point of intersection of that parallel with the diagonal rising from the sixth small division, we at last obtain a space corresponding to 365 feet, as was required. Had the quantity, instead of 365 feet, been 36 feet and 5 tenth parts of a foot, the distance would have been obtained in precisely the same way; for each of the great divisions would have been considered as representing not 100, but 10 units, each of the small divisions as one unit, and each of the diagonal advances would of course have represented tenth parts of an unit: and, on the other hand, had the great divisions been supposed to represent 1,000, each of the small divisions would have been 100, and each diagonal step would have been 10; consequently, on this supposition, the same opening of the compasses would have contained 3,650 feet.

This division of the unit into ten equal parts, is that commonly adopted for constructing scales, on account of the facility of computations in decimal arithmetic, when, by the mere removal to the right or left of the point which dis

tinguishes

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 ori ginal construction of which the difficulty they are intended to remove, must itself be previously surmounted.

Lines of equal parts, and of sincs, taugents, 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, sccant, &c. of any portion of the circumference may be ascertained, and registered in tables for the purposes of proportional calculation: but as calcu lations 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 arith metical 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.

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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 figure 1 over the 2; the second step, 2x2=4, is expressed by 2 over 4; the third step, 2x2x2=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 proportious between those in the lower row, or, in other words, the

upper

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