Page:The New International Encyclopædia 1st ed. v. 13.djvu/358

* MENSURABLE MUSIC. 324 MENTAL CONSTITUTION. Out of this system of notation our modern that the volume of : rectan^'iihir pariiUclepiped eystem of notation has been gradually evolved, or prism is found by nuiltiplying together the Those interested are referred to the comprehcn- length, breadth, and thiekness ; and of the oblique sive histories of music and cyclopaedias of Ger- parallelepiped, jirism. or cylinder, by multiply- ber, .Anibros, Coussemaker. See ilusiCAL NoTA- ing the area of the base by the height. TION. As in case of the circle, so in the mensuration MENSURATION (Lat. mensuratio, from of the cylinder, cone, and sphere, the theory of mensurare, to measure, from mensura, measure, limits (see Limit.s, Theory of) is applied in from mctiri, to njcasure). A branch of applied connection with the circumscribed and inscribed mathematics dealing with the calculation of linos, figures. The following formulas of mensuration angles, surfaces, and volumes from measured will be found convenient : Abbreviations; ft. base; 2j, altitude; r, radius: n. area; c, circumference; /), perimeter; «, slant height ; r, volume: m, mid-section; a. the number of radians in an angle. Parallelogram a ^=hh. Triiiugle a = > bh. Trapezoid a = (b--b') h. Parallelepiped v = bh. Prism V =bh. ]>ateral area, right prism « = ph. Prismatoid v ^ I h {b -- b' -- 4: m). Pyramid v = bh. Lateral area, regular pyramid o = i ps. Frustum of ])vramid v = I h (b -{- b' -- v 66'). Lateral area, frustum of regular pyramid a = 1 (p + p') s. Eight circular cylinder c = 6/i = ir rh. Lateral area a = ch — 2 r rh. Eight circular cone v = bh = J x r-h. Lateral area " = I cs ^ t is. Frustum of right circular cone v = ^ ir h (?y + ij^ + r, r,). Sphere r = |7r /■*, a = 4 ir /■-. Lune a = 2 a r-. Spherical polygon n = a r. Zone « ^ 2 TT rh. Spherical segment r ^ ^ ir /i [3 (r,' + r,') + h']. Splicrical sector v = I ir r- /i = J 6i-. Circle c ='2 v r, a = ir i^, arc^a ■ r. data. The metrical relations between lines and For the mensuration of geometric solids, con- angles are computed chiefly by the principles of suit Holzmiiller. Klemcnte der Steref S';"""^ O- ,,,,,,,,1., ^hi,!, .tv... to ^ive the unnd itsunity For the ijurposes of either direct measurement ,..,,.. ai . i »•*.,»;„;, or compu atimi a unit is necessarv. The straight •?"^, individual s.gmhcance. Mental const.tut on line is measured bv direct comparisons with scnne >* determined, irstot all, by the manner of the linear unit, as the inch, foot or yard. But in assembling of the elements which go to make up measuring a surface or a volume it is unneces- consciousness. Lvery normal mind comprises . '^ 1 ., „,,i.;„ ,..,;* r^,. manifold elements and diverging tendencies — sen- sary to applv an actual square or cubic unit, or v.- i • i i- ..i., even to d vide the magnitudes into such squares ^t';'"^- ""T-, vohtions-which oi.linaiily or cubes. It is onlv necessarv to measure el.rtain ^^'% ^•^^f'^ »« /""" ;.l%'.»'"t ,«'• '^^b.tual way^ of its boun.larv lines or dhnensiom. and from By dint of natural proclivities, due o .nheritanee these measurements to calculate the contents in "r environment, it achieves a kind of integuty terms of the a,,propriate unit; e.g. if a inches ■•""1 "■."l etfectiveness which we recognize as and 6 inches are the lengths of the adjacent sides poisonality : a mind is thus, as we say, well or- of a recti.ngle, its area is a-h- 1 square inch = «a>"=^eii operating to consistent and coherent n6 square inch; i.e. the number of scpuiie units -;"'.ls- Not infrequently, however, minds are de- of area in a rectangle is equal to the product "'''•'I't '" organization The weak-willed, mat- of two numbers which represent its ba.se and t^'t've person sufVers from lack of cohesion of altitude, measured bv the same linear unit. The mental elements: his interests vary with each areas of other figures are found from this by suggestion that conies to liim through ,H^rception the aid of C'ltain relations or properties of those "'• ."ily feeling: he is never cera in of Ins in_ figures; for instance, the area of a parallelo- teiitions, never constant in his attitude toward gram is the .same as the area of a rectangle hav- t''"'!-'^; never thoroughly self-possessed. On the ing the same base and altitude, and is therefore "ther hand, there are minds in which he niter- equal to the base multiplied bv the altitu.le. "al suggestion is so powerful as to dull I'cwp- ■s a triangle is half of a parallelogram of the tion to all not falling within a certain field of same base and the same altitude, its area is one- interest, .so destroying the mind's pliancy and half the product of its base and altitude. Certain powers of adaptation. Such minds have, we quadrilaterals and polvgons are measured by .say. strong jnepossessions; they are biased, nar- <lividing them into tri.'ingles. the area of each row; in extreme form they are afflicted with nf whii'li is separatelv calculated. ( Tor the ar.a fixed ideas and monomania. A third type of of the circle, see Circle.) By reasoning similar aberrant constitution is found in bi-ccntred or to that employed in the case of areas, it is shown multi-centred minds. Here the personality