Sturmius, Robertson, Marchetti, Hamilton, Emerson, Simpson, Bonnycastle, and Hutton; those of the three last are valuable and useful performances. Those who have written on the subject of practical Geometry, are Bayer, Bonnycastle, Clavius, Cantzlerus, Gregory, Herigon, Hawney, Hulsius, Kepler, Lightbody, Le Clerc, Mallet, Ozanam, Ramus, Reinhold, Schwinterus, Scheffelt, Tacquet, Voigtel, Wolfius, and many others.

ON THE USEFULNESS OF GEOMETRY.

No question is more frequently asked by beginners in Geometry, than the following: Of what use is the study of Euclid's Elements? The industrious, the idle, the sensible, and the dull, from different motives, are equally concerned in the inquiry: they almost daily agitate it with a degree of importunity, which sometimes proves troublesome to the Tutor, because he finds himself incapable of answering the question completely to his own or their satisfaction. The difficulty however lies not in the ignorance of the Tutor, or the want of usefulness in the science, but in the nature of things: for no art or science whatever can teach its own use; how then can one, who is learning merely the principles of Geometry, expect to understand fully its usefulness, or that his Tutor, however learned he may be, can by any explanation do justice to a science, of which the various and useful applications will perhaps never be completely determined? To try to satisfy all the absurd and vexatious scruples, which the idle, the querulous, or the captious, please to start against any branch of learning, would perhaps be a vain attempt; but it will be proper to advise the diligent and well-disposed student, (and to such the advice can hardly be needful,) that it is his duty, and will be to his advantage, to study attentively and without scruple, any branch of learning which his friends may think proper to recommend to him as useful, and which the experience of wise and good men in every age has proved to be so.

But in the present instance, an implicit reliance on authority is not at all necessary; the obvious uses of Geometry are suffi cient to recommend it to the candid and impartial inquirer; some of these we shall briefly enumerate. Geometry is useful, as it applies to the businesses and concerns of society, and as fundamental to other sciences and arts connected with them. Whatever relates to the comparison, estimation, &c. of distances, spaces, and bodies, belongs to Geometry; and consequently on its principles and conclusions immediately depend Mensuration, Surveying, Perspective, Architecture, Navigation, Fortification, with many other branches equally conducive to public benefit: in short, it is difficult to acquire a tolerable degree of know

ledge in philosophy, or any art or science, without some acquaintance with Geometry.

In addition to the direct and practical uses of the science, there is another, which Lord Bacon calls "collateral and intervenient." Geometry strengthens, corroborates, and otherwise improves the reasoning faculties, inuring the mind to patient labour, teaching it method, and supplying it with the means of contriving and adopting proper expedients for the prosecution of its researches. Geometry may then be justly considered as a highly valuable science, both with respect to its practical application, and as a complete model of strict demonstration: and in the latter view it recommends itself to the diligent attention of every lover of truth.

In what follows, we shall treat of Geometry in the two-fold view above explained, by briefly shewing the practical application of Euclid's doctrine, and likewise by considering it purely as a system of demonstration.

The demonstration of a proposition does not depend on the correctness of the diagram, which therefore may be drawn by hand; but in the practical uses of the propositions which we mean to exemplify, accurate figures should be made, and for this purpose instruments must be employed: we will therefore give a brief description of such instruments as are necessary for the construction of figures, and explain their farther uses hereafter, repeating, that the instruments are by no means necessary to the demonstration.

DESCRIPTION OF A CASE OF MATHEMATICAL INSTRUMENTS.

A common pocket case of Mathematical Instruments contains, 1. a pair of Plain Compasses; 2. a pair of Drawing Compasses; to the latter belong 3. a Port Crayon, 4. a Dotting Pen, and 5. a Steel Pen: 6. a Drawing Pen, with 7. a Pointer; S. a Protractor; 9. a Plain Scale; 10. a Sector; 11. a Parallel Ruler; and 12. a Black-lead Pencil *.

Cases of Mathematical Instruments may be had at all prices, from five shillings to six guineas; a case that costs twenty-five or thirty shillings will be suf

The PLAIN COMPASSES are used for the following purposes:

1. To draw a blank or obscure line by the edge of a ruler, through any given point or points.

2. To take the distance between two points, and apply it to any line or scale; or to take the length of one line, and apply it to another.

3. To measure any line by taking its length between the points of the compasses, and apply them to the divisions of a proper scale.

4. To set off any proposed distances on a given line. 5. To describe obscure circles, intersecting arcs, &c.

6. To lay off any proposed angle, and to measure a given angle, by means of a scale of chords, &c.

The DRAWING COMPASSES "; one of the legs is filled

ficiently good to answer the learner's purpose, and he should not go much under that price. A Magazine, or complete collection of every kind of useful drawing instrument, will cost from five to forty guineas.

In using the instruments, lines and figures should be drawn as fine, neat, and exact as possible; the paper on which the drawing is made should, if possible, not be pricked through or deeply scratched with the compasses; it should be laid on a quire of blotting, or other paper, during the operation; and the drawer should sit so that the light may be on his left, and not by any means in front. The drawing pen should not be dipped in the ink, but ink should be taken from the stand with a common pen, and put into it. The points of the instruments should be cleaned and wiped quite dry after they have been used, and every means employed to guard against rust, which will otherwise spoil the instruments.

b In the best sort of compasses, the pin or axle is made of steel, as also half the joint itself, as the opposite metals rubbing on each other are found to wear more equally; the points should be of hard well-polished steel, and the joint work with a smooth, easy, and uniform motion. In the drawing compasses, the shifting point is sometimes made with a joint, and furnished with a fine spring and screw; so that, having opened the compasses nearly to the required extent, by turning the screw the point will be moved to the true extent within a hair's breadth, for which reason they are named Hair Compasses.

There are various other kinds of compasses not appertaining to a common case of instruments, which are not less useful to the practical geometrician than those we have described; viz.

1. Bow Compasses, a small sort which shut up in a hoop; their use is to describe the circumferences and arcs of very small circles.

with a triangular socket and screw, to receive and fasten for use the following supplementary parts; viz. 1. a STEEL POINT; which being fixed in the socket, makes the compasses a plain pair, having all the uses above described. 2. A PORT CRAYON, with a short piece of black-lead or slate pencil, finely pointed and fitted on it for drawing circles and arcs on paper, or on a slate. 3. A STEEL PEN, for drawing lines or circles with ink; the small adjusting screw passing through the sides of the pen, serves to open or close them, for the purpose of drawing lines as thick or fine as may be thought necessary. 4. A DOTTER, which is a small indented wheel, fixed at the end of a common steel drawing pen; from which it receives ink for the purpose of drawing dotted lines or circles.

In the Port Crayon, Dotter, and Steel Pen, there is a joint for setting the lower part of the instrument perpendicular to the paper, which must be done in order to draw a line well.

The DRAWING PEN is fixed in a brass handle, and its use is to draw straight ink lines by the edge of a ruler. The handle or shaft unscrews near the middle, and in the end of the

2. Spring Compasses, or Dividers, made of hardened steel, having an arched head, which by its spring opens the legs; the opening being directed by a circular screw, and worked with a nut.

3. Proportional Compasses, both simple and compound; their uses are to divide a given line into any number of equal parts; to find the sides of similar planes or solids in any given ratio; to divide a circle into any number of equal parts, &c.

4. Trisecting Compasses, invented by M. Tarragon, for trisecting arcs and angles.

5. Triangular Compasses with three legs, for taking three points at once. 6. Turn-up Compasses are the plain compasses, with two additional points fixed near the bottom of the legs, the one carrying a port crayon, and the other a drawing pen; these are made with a joint to turn up, so as to be used or not, as occasion may require.

7. Beam Compasses for describing very large circles.

8. Elliptical Compasses for describing ellipses.

9. Spiral Compasses, for describing spirals.

10. Cylindrical and Spherical Compasses, or Calipers, for measuring the diameters of cylindrical and spherical bodies, &c. &c.

The Dotting Pen, not being easily cleaned, soon becomes rusty and useless; the best way to draw a dotted line is first to draw the line in pencil, and then to dot it with the writing or drawing pen.