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M E X

M E N

MFNSA, [Lilt a table,] in law-books, a term that includes inU all patrimony, and necessaries for livelihood.

MENSES , or CATAMENIA. See Medicne.

M ENSOULS, among the ancient Romans, harbingers, whosebusiness it was to go belore tire emperor, and fix upon lodgings forhim when lie travelled into any of the provinces.

M ks soni-:s, were uFn land-surveyors, architects, or appraisersof houses and public buildings.

•MENSTRUUM, in chemi-try, the fluid in which a solidbodv is dissolved. Thus w ater is a menstruum for salts and gums,and alcohol for resins.

MENSURATION , in general, denotes the act or art of mea-suring lines, superlicies, and soliils; and it is-, next to arithmetic, asubject oi the greatest use and importance, both in affairs that areabsolutely necessary in human 1 e, and in every branch of ma- ithematics ; a subject by which sciences are established, and com-merce is conducted ; by whose aid w e manage our business, andinform ourselves of the wonderful operations in nature ; by whichwe measure the heavens and the earth, estimate the capacities ofall vessels, and the bulks of all hollies, gunge our liquors, buildedifices, measure our lands, and the works of artificers, buy andsell an infinite variety of things necessary in life, and are suppliedwith the means of making die calculations which are necessary forthe construction of almost all machines.

Mensuration of Lines and Ancles, as applied to theDetermination of Heights and Distances.

Every magnitude is measured by a magnitude of the same kind,railed the measuring unit. Thus a line is measured by a line, anangle by an angle, a surface by a surface, and a solid by a solid.Certain magnitudes being given, that is, their measures being de-termined by an actual application of the measuring unit, it is thebusiness of mensuration to shew how the measures of others, whichdepend qn these may he obtained. By the mensuration and protrac-tion of lines and angles, the lengths, heights, depths, and distances,of objects are determined. Accessible lines are measured by ap-plying to them, some c ertain measure a number of times, as an inch,a toot, or yard ; but inaccessible lines must be determined by ameasurement of angles and accessible lines, by means of properinstruments, and the application of methods to be derived fromthe principles of geometry.

'1 he Instruments most commonly used for measuring heightsand distances are, a chain, a quadrant, a square, and a theodolite.

A Chain is used for measuring those distances, or lines, whichare to be given sides of triangles. The English chain is in length4 poles, or 6ti feet. It consists of 100 equal links made of iron,each link, therefore, sho.uld be 7 - 92 inches long, livery tenlinks, from one end to the middle of the chain, is distinguished bya mark made of brass.

A Quadrant is used for determining vertical angles; it isjuade of brass or wood, the radius being of any convenient length ;the circumference is divided into ninety equal parts, and theseagain subdivided as far as the dimensions of the quadrant will ad-mit. Also a plummet is suspended by a thread front the centre,and tw o sights fixed on one of the radii. See Plate CV. fig. 1.

A So. care, or Quadrant, is used for finding the proportion ofthe sides of a right-angled triangle. It is made of the same mate-rials. Two of its sides are divided into a hundred equal parts.This instrument is commonly called a geometrical square, Eig. 2.It is furnished with an index and a plummet.

A Theodolite is used for measuring horizontal as well as ver-tical angles. It is a circle of brass divided into degrees, &c, hav-ing an index moveable above its centre, and is furnished withsights. S.e Theodolite.

Erorlem 1. To find the height of an accessible object standingupon level ground.

I. By the quadrant, fig. j. Let any convenient distance, BA,lie mea. ured by the chain, in a direct line from the foot ot the per-pendicular, BC, that falls from the top of the object. Thenstanding at the point A, Id the quadrant be held as represented inthe figure, so that the eye at D may see the top of the object C,along die side of the quadrant, DE. • Now, if the plummet hangfreely, the. hue 1'P will be perpendicular to the horizon, andtiu.-retore parallel to B(J ; hence (lie angles DEP, DCK, are equal,and tiietr complements GKP, CDE, also equal. Thus ON, thearch of the quadrant that is remote from' the eye, will shew the

number of degrees in the angle ot elevation CDE. Whence, inthe right-angled triangle CK.D, the side DE (= AB), and ll*angle CDE being given, we may find CE by this proportion ; asradius to tiie tangent of CDE, so U DEto EC, to winch DA, theheight of the eye above the ground, being added, we get thewhole height of the object. If tiie angle of elevation be 43‘,then DE = EC ; that is, the distance measured is equal to theheight of the object above the eye.

II. By the square, fig. (j. Having measured AB as above, holdthe square to the eye D, as in the figure. Then, the plummethanging freely, the line EP cuts off from the square a - mall tri-angle similar to CDE. Therefore we shall have the proportion ofDid to EC ;and the former being given, the latter may be foundby the rule of proportion. Let n represent the number of equalparts which the plummet cuts otl'irotn the side Dll or I1G, to-wards the end D or G. Then, 1. When the plummet cuts tlu ;side CH remote from the eye, it is 100 : n : : DE : EC. lienee,if in this case, DE = 100, then EC = n. 2. When it passesthrough the opposite angle II, we have a ratio of equality, DE— EC. 3. When it cuts the side Dll, contiguous to the eve, his as n : 100 : : D"E : EC. And thus, by litis instrument the*height is found without the assistance of trigonometrical tables,

III. By means of two stalls, fig. 7. Let there be placed per*pendicularly in the ground, a longer staff’, DE, likewise a shorU'fone EG, so as the observer may see A, the top of the height 1°be measured, over the ends D, I' 1 , of the stalls; let I’ll aniPDC,parallel to the horizon, meet 1)E and AB in H and G : tlienth®triangles PHD, DCA, shall be equiangular; for the angles at Cand II are right ones: likewise the angle A is equal to FI)H !wherefore the remaining angles are also equal. Therefore as I'D >the distance of the two stall's, is to JID, the excess of the long 1 -’ 1 ',stall' above the shorter ; so is DC, the distance of the longer studfrom the tower, to CA, the excess of the height of the tower abovethe longer staff: anil thence CA will he found by the rule of three-To which if the length DE be added, you will have the wholeheight of the tower BA. 'Another method maybe occasionallycontrived for. measuring an accessible height, as bv the givenlength of the shttdow BD, fig. 8, I find out the height AB; fo r >letthere be erected a staff, CE, perpendicularly, producing 0®shadow EF: then it will be as El 1 ', the shadow of the staff, is t°EC, the staff itself; so is BD, the shadow of the tow er, to BA, tl' eheight. Though the; plane on which the shadow falls be not p 3 'rallcl to the horizon, if the staff be erected on the same plane, th erule w ill be the same.

IV. To measure an accessible height by means of a plain m ir *ror. Let AB, fig. 9, he the height to be measured ; let the mirro (be placed at C, in the horizontal plane BD, at a known distal'®®BC : let the observer go back to D, till he see the image of O®summit in the mirror, at a certain point of it, which he must dib'gently mark ; and let DE be the height of the observer’s ey®'The triangles ABC and KDC are equiangular; lor the angles 3tD and B are right angles; and ACB, ECD, are equal, being tbsangles of incidence and reileclion of the ray AC ; wherefore tl>®remaining angles at A and E are also equal. Therefore it will b®as C1.) is to DE, so is CB to BA.

A ntc 1. The observer will he more exact, if at the point P ®.staff be placed in the ground perpendicularly, over the top 0which the observer may see a point of the glass exactly in a ha®betwixt him and the tower.

A 'ole 2. In place of a mirror may he used the surface of ' va "ter, which naturally becomes parallel to the horizon,

'Flic angle C, tig. 10, being found by the geometrical quadra 11 ,’theodolite, Src. then in the triangle AliC, right-angled at B ,being supposed the horizontal distance of the observer front o ,l ltower) having the angle C, and the side BC, the required heig 1 !,will be found by plane trigonometry. Thus, suppose the anpJf '37° 24', and the horizontal distance, BC, 1 l(i, then the proporb 0will be as 11 ; T A. C :. Cl It : BA, the height.

The tangent altitude 37° 24' -— 9.8834103

Log. CB 110 — — — 2.0()44a3t>

Added I 1 t p478(iH3Radius 10.0000000

Height of the object AB 88.(3887 1.9478033

Supposing the observation lie niade on the tdp of the tower,

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