73 



MENISCUS. 



MENSURATION. 



674 



sensation. The muscles are either generally or partially paralysed, or 

 there may be spasmodic contractions. 



Should the attack continue, many of these symptoms may be 

 relieved, but the functions of the brain are imperfectly performed, and 

 the nervous and muscular functions are irregularly performed. 



The treatment of inflammation of the substance of the brain, like 

 that of inflammation of its membranes, must be determined by its 

 causes. If it arises from mechanical injury, an antiphlogistic system 

 must be pursued. If it accompanies the course of any disease pro- 

 duced by a morbid poison, the general symptoms must guide the 

 treatment of the local disease. Where it comes on from mental 

 distress, hard study, intemperance, or other causes, depletory measures 

 must be pursued with great caution. In a large number of these 

 cases a nutritious diet, with tonic medicines, are found to be most 

 beneficial. 



J1KXISCUS. [LENS.] 



MEXISPERMIXE (C U H,,NO ? ) is a white fusible crystallisable 

 alkaloid, forming salts with the acids found in the seeds of the Meni- 

 speiinum cocculus, known as C'occulus Indicia. 



MENSTRUUM. A term used hi chemistry to denote any liquid 

 which is employed as the solvent of a body. 



MENSURATION is the name given to a branch of the application 

 of arithmetic to geometry, which shows how to find any dimension of 

 a figure, or its area, or surface, or solidity, &e., by means of the most 

 simple measurements which the case will admit of. We need hardly 

 say that a complete treatise on this science would involve every branch 

 of mathematical science. We shall in this article collect together the 

 most important rules, the method of using which will be obvious to 

 all who can employ the trigonometrical tables. By the length of a 

 line we mean the number of linear units contained in it, and by its 

 (square and cube the number of units multiplied by itself once and 

 twice. 



The measurement of lengths and directions resolves itself for the 

 most part into the determination of a side or angle of a triangle, when 

 other sides or angles are given. The triangle may be either on a plane 

 or on a sphere ; but we refer the latter to SPHERE, since the use of 

 spherical trigonometry can only be well explained in connection with 

 astronomy. Let a, b, c be the sides of a triangle, and A, B, c the 

 opposite angles. If the triangle be right angled at c, we have the 

 following formula; : 



ac sin A=C cos u = 4 tan A = 6 cot B 

 4 =c sin B = C cos A = U tan B=acot A 



a 



sin A 



a 

 cos B 



6 b 



~ sin B ~~ cos A 



cb) b= V(c + a.c o) 



The preceding formula) contain the solution of every case of right- 

 angled triangles. 



\Ve now pass to oblique-angled triangles, of which there are four 



1. Given the three sides, o, b, and c, to find the angles. Let the 

 perpendicular let fall from c upon the longest side c divide it into 

 two segment* a and adjacent to a and b, and let b be 7 o. Then the 

 equations 



(b-a)(b+a) 

 /3 + a=c, /8-o= - - 



(in which /3 a is easily found by logarithms), will give j8 and a. 

 Then 



a 8 



00B= - COSA=-j C=180 ( '-(A + B) 



Another method is as follows. Compute M from the following,* 



a+b + e //> a -a &. c 



M= 



2. Given two sides a and J, and the remaining angle c, required 

 0, A, and B. Firstly, to And the angles, determine 



4 (8 + A) from 4(B+A)=4(180-o) 



b-a 

 (B A) from tan J (B-A) = j . cot 4 o 



A=J(B + A)-4(B-A); B=4(B + A) + 4(B-A) 



sin c .in o 



cb - a, . 



sin B sin A 



* Tbti conwnlent adaptation of a well-known formula is found, we believe, 

 for the Drat time in Frofcwor Wallace'! work on Mathematical Theorems ami. 

 Formula) (Longman, 1839). 



To find the third side without the aid of the angles, assume 



V(" V) 2 cos -5 o 



-- , then c=(o+i) cos 6; 



V(a6).2sin4o a-5 



tan *= -- = -- then c = 



3. Given a, 6, and the angle A, to find the rest : 



b a sin o 



smB= - sin A, c = 180 (A+B). c= . 

 a sm A 



When a< b, B may be taken either acute or obtuse ; and the problem 

 has two solutions. 



4. Given a, and two of the angles, to find the rest. It is unneces- 

 sary to distinguish the angles given, as two immediately determine 

 the third. 



sin B sin a 



' =a iIn"7' C=a s!n~I 

 The area of the triangle is 



a b sin c be sin A ca sin B 

 2 - or - 2 - or - 2 - r V\? ' s ~ a *~* s ~ c ) 



The perpendiculars let fall from the vertices A, B, and c, upon the 

 opposite sides, are severally 2 V(a 3 a .8 b. ac) divided by 0,6, 

 and c. 



1 > i , i m. of inscribed circle 2 V(a a A s c-f- s) 



a, 6 c 



Do. of circumscribed circle - - or - - or - 

 sin A sin B sm o 



or abc-~2 V(-a-a .a 6.a c) 

 Segments of c made by perpendicular from c, 



i* 



Adjacent to a, 

 Segments of c, by line bisecting o, 



; to b, 



2o ' 



ac be 



Adjacent to o, T ; to J, r. 



^. 

 Line bisecting c= --- r 



c) 2ab 



= , cos 4 o 



Line bisecting c = 4 V(2 a" + 2 6 c'). 



The area of a rectangle (in square units), and that of a parallelogram, 

 is the product of the units in the base and perpendicular distance of 

 the opposite sides. But if two sides only be parallel, half the sum of 

 the parallel sides must be multiplied by the perpendicular distance 

 between them. In other cases, the figure must bo measured by 

 dividing it into triangles, except when it is either a four-sided figure 

 capable of inscription in a circle, or a regular polygon. Every triangle 

 is half of the rectangle contained by any one of its sides, and the 

 perpendicular let fall from the opposite vertex. 



If a, b, c, and d be the sides of a four-sided figure inscribed in a 

 curie, and a their half-sum, the area is 



V(a <*& ,8e.t d). 



If a be one of the sides of a regular polygon of n sides, the area of 

 the figure and the diameters of the circumscribed and inscribed circles 

 are 



cot 



180 



180" 



/180 



' and - f - tan ~ 



Tables connected with this subject are given in the article REGULAR 

 FIGURES. For the method of measuring irregular areas, see QUAD- 

 RATURES, METHOD OF. 



The whole of the measurement of the circle depends upon the ratio 

 of the circumference to the diameter, which is called IT, and is 

 3-1415927 very nearly, or ! , 2 roughly, or Jf| very nearly. [ANGLE.] 

 So many simple derivations from this number are practically useful, 

 that we shall give a table of them, accompanied by their logarithms, 

 first giving a method of multiplying and dividing by IT, which is a 

 correction of the use of *f. To multiply by TT, multiply by 22 and 

 divide by 7 ; from the result take one-eighth of the hundredth part of 

 the multiplicand as a correction ; the result is too great only by about 

 its 200,000th part. To divide by TT, multiply by 7, divide by 11 and 2, 

 and to the result add the eighth part of the thousandth part of 

 the dividend; the result is too small by very nearly its 100,000th 

 part. 



