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4thly. To shew that π=



The history of this remarkable approximation to the value of π is involved in some obscurity. It is certainly due to Peter Metius; for his son, Adrian Metius, in his Geom. Pract. (A.D. 1640), says that his father published this ratio in answer to the quadrature of Simon à Quercii, supposed to be Simon Duchesne; and it was done, he says, Archimedeis demonstrationibus, meaning, probably, by the inscribing &c. of polygons. Nothing further seems to be known respecting the ratio. But Professor De Morgan, a high authority on such subjects, has kindly furnished me with a clever conjecture of his own as to the probable method employed by Peter Metius. He thinks it likely that was only an appendix to Metius' result from the polygons, and not the result itself. That result would be 3.14159265; and




having this before him, and moreover knowing that 1


and are limits between which lies, he tried the


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therefore is correct to 6 places of decimals, and no further.

It is also easily retained in the memory from the circumstance that it is composed of the first three odd numbers in pairs, 113 355, taking the first three digits for the denominator and the remaining three for the nume


* The same result may now be easily obtained, by the method of 31415926 'Continued Fractions,' from But this method was not in


use in the days of Peter Metius.

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244. When the linear dimensions of any surface have been obtained, and it is required to make a representation thereof; or, when the dimensions of any diagram of a surface have been obtained, and it is required to make another diagram of dimensions either larger or smaller than those of the original; it is clearly necessary to adopt some means whereby we may be sure that, whatever size we determine upon, the magnitudes of all the lines of the representation or diagram, which we are about to make, may bear a certain uniform ratio to those of the corresponding lines in the original.

This process is termed drawing to a Scale. For this purpose the draughtsman either divides for himself a straight line on paper into such parts as will best suit his purpose, or procures an instrument so divided, that from it the various dimensions which have once been obtained by actual measurement, may be accurately transferred to his plan, according to some fixed proportion previously agreed upon.

Such an instrument is called a Scale. And it is usual in every such diagram, map, or plan, either to express in the margin the proportion which every line, or length, in it bears to some stated unit, or actually to draw at the foot of the diagram, map, or plan, the scale according to which it is constructed.

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Thus, if it be written in the margin of a diagram, map, or plan, Scale, a yard to one-tenth of an inch,' then every line, or length, in the diagram, map, or plan, which is


measured by th of an inch, actually means 1 yard;






ths of an inch mean 2 yards; ths 3 yards; and so on. In this case we require both compasses, and the Plain Scale described below. But if the Scale itself be drawn at the foot of the diagram, map, or plan, then the compasses only are required, to enable us either to determine the length of any line already drawn, or to draw other lines in strict proportion.

245. A PLAIN SCALE, when it assumes the form of an instrument, consists of a thin flat rectangular piece of box


or ivory*, usually about 6 inches long, and 13 inches broad; and in its simplest form contains only two parallel lines on one of its faces, drawn in the direction of its length, and divided by small lines at right angles to the former, and at equal intervals of 1 inch, in., or some other unit of length agreed upon. Thus ABDC represents such an instrument, the two parallel lines on its face being divided into 6 equal parts, and the points of division marked 0, 1, 2, 3, 4, 5.

987654321 0


The length of each of the portions so formed may be taken to represent one mile, or one yard, or other unit of length; and for the subdivision of the unit the first of them to the left is divided into so many equal parts, that each shall represent one of the denomination next inferior to that of the assumed unit, or some convenient number of them. Thus, if the Scale be one of feet, the subdivisions will be inches, that is, twelfths of the unit: if the Scale be one of yards, the subdivisions will be feet, or thirds of the unit; if of miles, the subdivisions will be furlongs, or eighths of the unit, &c. For general purposes, however, it is most convenient to subdivide the unit into tenths, as in our diagram above.

When this Scale is used, and the unit on it is an inch, as for example in laying down lines, or lengths, to a scale of a yard to an inch, suppose we want to find the line corresponding to 48 yards; put one foot of the compasses upon the point in the Scale numbered 4, and the other upon the number 8 in the subdivisions of the unit; then it is clear, that there is intercepted between the points of the compasses a length of 4 units and 8 tenths, that is, 48. And by considering the unit on the

* Ivory, though commonly used, is a bad material for the purpose, since its length varies with moisture; box is much better.



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cepted length will represent 4-8, or 48, or 480, or •48, or 048, respectively.

Ex. 1. Represent 118 ft. in a Scale of a foot to onetenth of an inch.

This quantity will measure on the Scale 118 tenths of an inch, or 11.8 inches, i. e. once the length of the Scale, (if its whole length be 6 inches,) and 5.8 inches more. So that, after taking the whole length of the Scale by the compasses, a second measurement must be taken, wherein one point will be exactly at the end of the Scale, on the figure 5, and the other upon the subdivision marked 8.

Ex. 2. What must one inch of the Scale represent, in order that 18 feet may be represented upon it, by the distance between the large division marked 1, and the subdivision marked 8?


Here 18 ft.=


8 th of 1 foot + 100

ths of 1 foot; hence,


each of the large divisions must represent th of 1 foot,



and the small divisions th of 1 foot; or, the Scale will


be one-tenth of a foot to an inch, or 1 foot to 10 inches.

If the quantity had been 018 feet, the Scale must be 1 inch to one-hundredth of a foot, or 100 inches to a foot.

N.B. The subdivisions of the unit may be other than


Thus, suppose them to be twelfths of an inch, each twelfth representing 1 foot; and let it be required to measure 43 ft. thereby.

Then, 43 ft.=43 twelfths of an inch-3 inches, hence the compasses must embrace 3 units and 7 twelfths; or, one point must be upon the larger division marked 3, and the other on the subdivision marked 7.

Nothing less than 1 foot could be laid down from this Scale. Also any large number of yards and feet,

as 93 yds. 2 ft., must be converted into feet, viz. 281 feet, and this would give on the Scale 281 twelfths, or 23 in. And, since the Scale embraces only 6 units, or inches, the length 23 in. would be laid down by repeating the whole length of the Scale 3 times, and then taking 52 more, as the 3 was taken above.

But if it is needful to carry the division to hundredths of an inch, so as to lay down a line whose measure consists of three places of figures, as 487, the above tenths must be further divided, each into tenths, or the unit into hundredths.

But if such divisions were made, few could count them. To enable us to make use of these minute subdivisions without confusion, we construct, or procure, what is called a


246. It has been shewn in (172, Part 11.) how to

take any required portion as



th, ths, &c....... of a 10 10

small straight line; and if the small line be itself a tenth of any assumed unit, as 1 inch, then the tenths thereof will be hundredths of the same unit.

Thus, let ab, in the annexed diagram, be the tenth of the unit; from b draw an indefinite straight line at right angles to ab, and with any small opening of the compasses step along this line from b to c, dividing bc into 10 equal parts, and marking the points of division 1, 2, 3, 4, 5, 6, 7, 8, 9, as in the diagram.

Join ac, and through the points of division let lines be drawn parallel to ab; these parallels intercepted between ac, and bc, beginning with the least, will therefore be

1 2 3




&c. of ab,

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100' 100' 100'

of the unit.



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