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# Scale

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SCALE (Mathematics). A scale is any line drawn upon wood or other solid substance, and divided into parts, equal or unequal, the lengths of which may be taken off by the compasses, and transferred to paper, in aid of any geometrical construction. The manner in which the scale is divided depends of course upon the nature of the alge braical or trigonometrical expression the ambles of which are to be represented. When the subdivisions of a scale are equal, any of the methods noticed in GRADUATION may be employed to obtain them ; but in other cases, and indeed in the preceding one, it is usual to form scales by copying from an original which is carefully made in the first instance.

The most simple of all scales is that in which the sub divisions are all equal, or, as it is called, a scale of equal parts. Such a scale is not only the most easily constructed, but may be considered as containing all other scales. For example, suppose it required to lay down very accurately an angle of 25°. It appears [Cuonn] that if the radius contain 500 equal parts, the chord of 25° contains 216 such parts and of a part. With a good scale of equal parts, and 500 of them taken as a radius, the angle may be laid down, if required, much more correctly thau by a common scale of chords. [PROTRACTOR.] The largest table of chords which is laid down on common scales has a radius of 3 inches, the 500th part of which, or about the 167th of an inch, is a very small length ; and it is difficult to trust any scale so far without verification, except the scale of equal parts. In the latter species, one part may be tried against another. and any one may for himself very soon ascertain whether there ho any perceptible error.

In all the moat accurate species of drawing, it is better to rely on tables and a really good scale of equal parts than on any of the common scales, though the latter are generally very good, and will do abundantly well for onlinary purposes.

Long scales of equal parts are made with different aulxliviiiions, ranging from the 30th part of an inch to the 50th. If the substance of the scale be ivory, an inch will very well bear diviaiou into 60 parts, but 50 is more convenient for decimal calculation. A common ivory scale, of a rectangular form, such as is usually found in cases of drawing-instruments, if it have no trigunometneal lines laid down, usually contains the following scales of equal parts: I. The quarter of an inch divided into 10 equal parts, each of which is again subdivided into 10 equal parts by a DIAGONAL &ALL There are commonly two diagonal scales, one at each end of the scale of quarters, the one on the left dividing the Sth of an inch into 100 parts, and the ,one on the right the quarter. It will easily be seen that the 400th of an inch is a uselessly small quantity, even when the lines are drawn on ivory.

2. A set of scales in which the inch is severally divided into 30, 35, 40, 45, 50, and 60 equal parts : 10 of these ]arts make, in each case, one of the larger subdivisions of the scale, and one larger division is also divided into 12 equal parts ; so that, when the larger division is made to represent a foot, feet and inches may be easily laid down.

3. A set of scales in which the larger divisions are 4, :„ I, 1, ;, and ; of an inch. The larger division is, as before, divided both into 10 and 12 parts.

When trigonometrical lines are laid down they are usually one or two scales of chords, the radius of each of which is found by its chord of 60 degrees; is scale of rumba, which is nothing more than a scale of chords, the angular unit being, not a degree, but a point of the compass ; a scale of silica, with one of secants sometimes added ; a scale of tangents, and of scmitangents, the latter being really the same scale as the former, but marked with double angles, scruitangent being a technical term, not for the half of a tangent, but for the tangent of half an angle. We shall have something more to say of these lines under SECTOR. In Gunter's scale, as it is called, which is a seek of 2 feet in length, used in navigation, there aro also scales of logarithms, of numbers, sines, tangents, &c., and also a scale of meridional parts [Rena Lnee); of these logarithmic scales we shall have to speak more particularly under SLIDINU RUM SCALE (5Iusic). A great deal has been written on this subject, IT mathematicians, by musicians, and by those who combine both characters ; but, from various circumstances, hardly anything which is accessible to the young arithmetician wishing for something which may really he a help to him iu his musical studies. The Greek scale ;Mum; Ts-reaseiloile], the only fruitless subject of inquiry out of all that is Greek, has exhausted the learning, science, and ingenuity of the best writers, with no result but this, that over-refinements of theory are found either to have hindered practical excellence, or to have arisen out of the want of it ; most likely the latter. The learning however which it was necessary to apply to the explanation of the Greek writers, has made it usual to write on this subject more pro foundly than on others of the same difficulty : it is our object in the present article to explain the musical scale, if possible, more simply, and in its simplest parts : leaving to the article TLIIITRA.IIENT such considerations as, arising out of the present article, are required by those who would understand the higher practical details of the subject.

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