Chav. VIII.

GEOMETRY.

807

regarded, and maintained, as the slightest inclination would affect the truth of the observa-tions made with it.

To measure by means of a Piquet and the Puleof Three a Height of which the Foot is accessible. —

Place on the ground at some distance a piquetCE, of any length ;*then retire from this piquettill, by lowering the eye to the ground, as atD, the top of the piquet and the summit of theheight, B, to be measured, is seen in the samevisual ray.

State the question by placing in the first termthe distance from the point where the eye wasplaced when on the ground from the piquet; in thesecond term, the distance from the same point tothe base of the required height, and for the thirdterm, the height of the upright piquet: the quo-tient will be the required height.

From D C we will suppose 5 feet, and from I)to A 25 feet, and the height of the piquet, C E, 6feet: then we shall have

5 : 25 : : 6 : 30 feet for the height of A B.

For if a line be drawn in a triangle parallel to oneof its sides, it will cut the two other sides propor-tionally ; and the line which bisects any angle ofa triangle divides the opposite sides into two seg- /

ments which are proportional to two other adja-cent sides: let the angle DEA of the triangleDAE be bisected by the line CE, making thetwo angles at E so bisected equal : then the segment DC will be to the segment C A, asthe side D E is to the side E A, or D C will be to C A, as PE is to E A, and the line E Cbeing drawn perpendicular or parallel to B A, cuts the two other sides proportionally,making D C to C A, as is D E to E B, or to its equal E A.

When the sun shines, if we set up vertically a staff of any known length, and measurethe length of its shadow upon a horizontal or other plane, and measure also the length ofthe shadow of the object whose height is required, we may, by a similar rule, obtain it; asthe length of the shadow of the staff is to the length of the staff itself, so is the length ofthe shadow of the object to the object’s height.

To draw the Map of a Country d la Cavaliere. — We must be placed on an eminencewhich gives an opportunity of seeing the country to be mapped, and have some attendantacquainted with the names of all the places and objects before and around, as well as theirdistances from each other, that their relative positions may be set down. To transfer thisrough map, or to draw it out fair, we must begin by drawing a line from the top to thebottom, and on this form a scale divided into as many miles as there are between the mostdistant places; then mark the site of the different places which come on this line, by takingin the compasses the distance right or left from the first position, performing the sameoperation with all the others : rivers may then be traced, as well as the various objects betweenthem.

It would be advisable in mapping a country ahvflys to work upon a meridian line, andbefore any survey is commenced, its direction should be accurately laid down : where anumber of parishes are surveyed, it is important that general instructions should be im-plicitly followed, that the whole when brought together may be examined : for example,all writing and figuring should be placed in the same direction ; the top of the paper onwhich the representation is made should be considered north, and whatever the form of theplan, this rule should never be departed from : upon a sketch so made, a series of trianglesmay be extended at any time, and the respective sides calculated with precision. From aneminence or lofty site numerous towns and villages may be seen, and after the meridianline is established, their bearings from each other may be noticed upon the divisions of acard, which if drawn within a circle will serve to make the sketch. From a table showingthe distance in miles of one place from another, this may be rectified and brought to ap-proach the truth : or a circle may be traced on the ground, and after dividing it into3GO degrees, piquets may be set upon that part of its circumference which is in a line withthe object, and the observer being stationed in the centre of the figure, the divisions of thecircle may be noticed, and thus the sketch of a country may be made where instrumentscannot readily be obtained, or a hasty survey is to be laid down.

If an estate could be seen at once, and its meridian line be determined, there would beno difficulty in mapping it, and approximating its area, by walking across it in the longestdirection, afterwards triangulating it from this line as a base, and then uniting ft with th»

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