On the lines, make the lateral distance 10, a distance between eight on one leg, and six on the other leg.

On the sines, make the lateral distance 90, a transverse distance from 45 to 45; or from 40 to 50; or from 30 to 60; or from the sine of any degrees to their complement.

Or on the sines, make the lateral distance of 45 a transverse distance between 30 and 30.

SELECT GEOMETRICAL PROBLEMS.

Science may suppose, and the mind conceive things as possible, and easy to be effected, in which art and practice often find insuperable difficulties. "Pure science has to do only with ideas; but in the application of its laws to the use of life, we are constrained to submit to the imperfections of matter and the influence of accident." Thus practical geometry shews how to perform what theory supposes; in the theory, however, it is sufficient to have only a right conception of the objects on which the demonstrations are founded; drawings or delineations being of no further use than to assist the imagination, and lessen the exertions of the mind. But in practical geometry, we not only consider the things as possible to be effected, but are to teach the ways, the instruments, and the operations by which they may be actually performed. It is not sufficient here to shew, that a right line may be drawn between two points, or a circle described about a centre, and then demonstrate their properties; but we must actually delineate them, and exhibit the figures to the senses: and it will be found, that the drawing of a straight line, which occurs in all geometrical operations, and which in theory is conceived as easy to be effected, is in practice attended with considerable difficulties.

To draw a straight line between two points upon

a plane, we lay a rule so that the straight edge thereof may just pass by the two points; then moving a fine pointed needle, or drawing pen, along this edge, we draw a line from one point to the other, which for common purposes is sufficiently exact; but where great accuracy is required, it will be found extremely difficult to lay the rule equally, with respect to both the points, so as not to be nearer to one point than the other. It is difficult also so to carry the needle or pen, that it shall neither incline more to one side than the other of the rule; and thirdly, it is very difficult to find a rule that shall be perfectly straight. If the two points be very far distant, it is almost impossible to draw the line with accuracy and exactness; a circular line may be described more easily, and more exactly, than a straight, or any other line, though even then many difficulties occur, when the circle is required to be of a large radius.

It is from a thorough consideration of these difficulties, that geometricians will not allow those lines to be geometrical, which in their description require the sliding of a point along the edge of a rule, as in the ellipse, and several other curve lines, whose properties have been as fully investigated, and as clearly demonstrated, as those of the circle.

From hence also we may deduce some of those maxims which have been introduced into practice by Bird and Smeaton, which will be seen in their proper place. 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 observation, 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

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made and divided; and that these sciences will advance in proportion as those are less difficult in their use, and more perfect in the performance of their respective operations.

There is scarce any thing which proves more clearly the distinction between mind and body, and the superiority of the one over the other, than a reflection on the rigid exactness of speculative geometry, and the inaccuracy of practice, that is not directed by theory on one hand, and its approximation to perfection on the other, when guided by a just theory.

In theory, most figures may be measured to an almost infinite exactness, yet nothing can be more inaccurate and gross, than the ordinary methods of mensuration; but an intelligent practice finds a medium, and corrects the imperfection of our mechanical organs, by the resources of the mind. If we were more perfect, there would be less room for the exertions of our mind, and our knowledge would

be less.

If it had been easy to measure all things with exactness, we should have been ignorant of many curious properties in numbers, and been deprived of the advantages we derive from logarithmns, sines, tangents, &c. If practice were perfect, it is doubtful whether we should have ever been in possession of theory.

We sometimes consider with a kind of envy, the mechanical perfection and exactness that is to be found in the works of some animals; but this perfection which does honour to the Creator, does little to them; they are so perfect, only because they are beasts.

The imperfection of our organs is abundantly made up by the perfection of the mind, of which we are ourselves to be the artificers.

If any wish to see the difficulties of rendering

practice as perfect as theory, and the wonderful resources of the mind, in order to attain this degree of perfection, let him consider the operations of General Roy, at Hounslow-heath; operations that cannot be too much considered, nor too much praised by every practitioner in the art of geometry. See Philosoph. Trans. vol. 80, et seq.

PROBLEM 1. To erect a perpendicular at or near the end of a given right line, CD, fig. 5, plate 4.

Method 1. On C, with the radius C D, describe a faint arc ef on D; with the same radius, cross ef at G; on G as a centre, with the same radius, describe the arc DEF; set off the extent DG twice, that is from D to E, and from E to F. Join the points D and F by a right line, and it will be the perpendicular required.

Method 2, fig. 5. On any point G, with the radius DG, describe an arc FED: then a rule laid on C and G, will cut this arc in F, a line join. ing the points F and D will be the required perpendicular.

Method 3. 1. From the point C, fig. 6, plate 4, with any radius describe the arc rnm, cutting the line A C in r. 2. From the point r, with the same radius, cross the arc in n, and from the point n, cross it in m. 3. From the points n and m, with the same, or any other radius, describe two arcs cutting each other in S. 4. Through the points S and C, draw the line S C, and it will be the perpendicular required.

Method A. By the line of lines on the sector, fig. 7, plate 4. 1. Take the extent of the given line A C, 2. Open the sector, till this extent is a transverse distance retween 8 and 8. 3. Take out the trans verse distance between 6 and 6, and from A with that extent sweep a faint are at B. 4. Take out the distance between 10 and 10, and with it from C, cross the former arc at B. 5. A line drawn

through A and B, will be the perpendicular required; the numbers 6, 8, 10, are used as multiples of 3, 4, 5.

By this method, a perpendicular may be easily and accurately erected on the ground.

Method 5. Let A C, fig, 7, plate 4, be the given line, and A the given point. 1. At any point D, with the radius D A, describe the arc EAB. 2. With a rule on E and D, cross this arc at B. 3. Through A and B draw a right line, and it will be the required perpendicular.

PROBLEM 2. To raise a perpendicular from the middle, or any other point G, of a given line A B, fig. 8, plate 4.

1. On G, with any convenient distance within the limits of the line, mark or set off the points n and m. 2. From n and m with any radius greater than GA, describe two arcs intersecting at C. 3. Join CG by a line, and it will be the perpendicular required.

PROBLEM 3. From a given point C, fig. 8, plate 4, out of a given line AB, to let fall a perpendicular.

When the point is nearly opposite to the middle of the line, this problem is the converse of the preceding one. Therefore, 1. From C, with any radius, describe the arc n m, 2. From mn, with the same, or any other radius, describe two arcs intersecting each other at S. 3. Through the points CS draw the line CS, which will be the required line.

When the point is nearly opposite to the end of the line, it is the converse of Method 5, Problem 1, fig. 7, plate 4.

1. Draw a faint line through B, and any conve nient point E, of the line A Č. 2. Bisect this line at D. 3. From D with the radius DE, describe an arc cutting A Cat A. 4. Through A and B