many difficulties occur, when the circle is required to be of a large radius.

"And let no one consider these reflections as the effect of too scrupulous exactness, or as an unnecessary aim at precision; for, as the foundation of all our knowledge in geography, navigation, and astronomy, is built on observations, and all observations are made with instruments, it follows that the truth of the observations, and the accuracy of the deductions therefrom, will principally depend on the exactness with which the instruments are made and divided, and that those sciences will advance in proportion as these are less difficult in their use, and more perfect in the performance of their respective operations.'

OF PARALLEL RULES.

Parallel lines occur so continually in every species of mathematical drawing, that it is no wonder so many instruments have been contrived to delineate them with more expedition than could be effected by the general geometrical methods of the various contrivances for this purpose, the following are those most approved,

The Common Parallel Rule.-This consists of two straight rules, which are connected together, and always maintained in a parallel position by the two equal and parallel bars, which move very freely on their centres, or rivets, by which they are fastened to the rules.

The Double Parallel Rule.-This instrument is constructed exactly upon the same principles as the foregoing, but with this advantage, that in using it, the movable rule may always

be so placed, that its ends may be exactly over, or even with the ends of the fixed rule, whereas in the former kind, they are always shifting away from the ends of the fixed rule; it consists of two equal flat rules, and a middle piece; they are connected together by four brass bars, the ends of two bars are rivetted on the middle line of one of the straight rules; the ends of the other two bars are rivetted on the middle line of the other straight rule; the other ends of the brass bars are taken two and two, and rivetted on the middle piece, as is evident from the figure. It would be needless to observe, that the brass bars move freely on their rivets, as so many centres.

The Cross-barred Parallel Rule.-In this, two straight rules are joined by two brass bars, which cross each other, and turn on their intersection as on a centre; one end of each bar moves on a centre, the other slides in a groove, as the rules recede from each other.

As the three parallel rules above described are all used in the same way, one problem will serve for them all.

A right line being given, to draw a line parallel thereto by either of the foregoing instruments :

Set the edge of the uppermost rule to the given line; press the edge of the lower rule tight to the paper with one hand, and with the other move the upper rule, till its edge coincides with the given point; and a line drawn along the edge through the point is the line required.

The Rolling Parallel Rule, is a rectangular parallelogram of black ebony or brass. It is supported by two small wheels, which are connected together by a long axis, the wheels being exactly of the same size, and their rolling surfaces being parallel to the axis. When they are rolled backwards or forwards, the axis and rule will move in a direction parallel to themselves.

The wheels are somewhat indented, to prevent their sliding on the paper. In rolling these rules one hand only must be used, and the fingers should be placed nearly in the middle of the rule that one end may not have a tendency to move faster than the other. The wheels only should touch the paper when the rule is moving, and the surface of the paper should be smooth and flat. This rule is sometimes made with slips of ivory laid on the edges and divided into inches and tenths.

CHAPTER VIII.

ON SCALES.

SCALES of equal parts are used for measuring straight lines, and laying down distances, each part answering for one foot, one yard, one chain, &c., as may be convenient, and the plan will be larger or smaller as the scale contains a smaller or a greater number of parts in an inch.

Scales of equal parts may be divided into three kinds; simply divided scales, diagonal scales, and vernier scales.

Simply divided Scales.-Simply divided scales consist of any extent of equal divisions, which are numbered 1, 2, 3, &c., beginning from the second division on the left hand. The first of these primary divisions is subdivided into ten equal parts, and from these last divisions the scale is named. Thus it is called a scale of 30, when 30 of these small parts are equal to one inch. If, then, these subdivisions be taken as units, each to represent one mile, for instance, or one chain, or one foot, &c., the primary divisions will be so many tens of miles, or of chains, or of feet, &c.; if the subdivisions are taken as tens, the primary divisions will be hundreds; and, if the primary divisions be units, the subdivisions will be tenths.

The above diagram represents six of the simply divided scales, which are generally placed upon the plain scale. To adapt them to feet and inches, the first primary division is divided duodecimally upon an upper line. To lay down 360, or 36, or 3.6, &c., from any one of these scales, extend the compasses from the primary division numbered 3 to the 6th lower subdivision, reckoning backwards, or towards the left hand. To take off any number of feet and inches, 6 feet 7 inches for instance, extend the compasses from the primary division numbered 6, to the 7th upper subdivision, reckoning backwards, as before.

Diagonal Scales.-In the simply divided scales one of the primary divisions is subdivided only into ten equal parts, and the parts of any distance which are less than tenths of a primary division cannot be accurately taken off from them; but, by means of a diagonal scale, the parts of any distance which are the hundredths of the primary divisions are correctly indicated, as will easily be understood from its construction, which we proceed to describe.

Draw eleven parallel equidistant lines; divide the upper of these lines into equal parts of the intended length of the primary divisions; and through each of these divisions draw perpendicular lines, cutting all the eleven parallels, and number these primary divisions, 1, 2, 3, &c., beginning from the second.

Subdivide the first of these primary divisions into ten equal parts, both upon the highest and lowest of the eleven parallel lines, and let these subdivisions be reckoned in the opposite direction to the primary divisions, as in the simply divided scales.