1 
 
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 OF ILLINOIS 
 LIBRARY 
 
 5l0.2iz 
 
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 Latest Date stamped below. A 
 charge is made on all overdue 
 books. 
 
 University of Illinois Library 
 
 
 M32 
 
Stewart's Educational Series. 
 
 ‘WEINKLES’ 
 
 IN 
 
 ALGEBRA, ARITHMETIC, EUCLID, 
 MECHANICS, CHEMISTRY, HEAT AND LIGHT, 
 MAGNETISM AND ELECTRICITY, 
 
 BEING 
 
 FORMULAE, DEFINITIONS, LAWS AND PROPOSITIONS 
 
 FOR THE USE OF 
 
 LONDON MATRICULATION AND OTHER STUDENTS 
 
 ARTHUR G. MADGE, B.A. 
 
 \t 
 
 LONDON: 
 
 W. STEWART & CO., Ld., HOLBORN VIADUCT STEPS, E.C. 
 

 
 
 
 
 
 
 
 
 
$ig /(4J-/ZZ SI ••*«*»» 
 
 PREFACE. 
 
 This little book pretends to be nothing more than a 
 
 summary of Formulae, &c., for the use of Matriculation 
 students, who, after reading, require to remember accurately 
 and thoroughly the Propositions, Theorems, &c., they have 
 read. 
 
 Each section should be revised frequently, and the book 
 being so small, can easily be carried in one’s pocket and used 
 whenever, amidst other work, there may be a few minutes' 
 leisure. By using the book daily in this way, the student 
 will soon thoroughly know the skeleton of his work, and 
 then in his later revisions will be able, with his additional 
 information, to clothe it. 
 
 ARTHUR G. MADGE, B.A., 
 Endowed School, 
 
 Watford, Herts. 
 
CONTENTS. 
 
 44 
 
 Section 
 
 I. 
 
 Algebra, 
 
 • 
 
 Section 
 
 II. 
 
 Arithmetic, 
 
 • 
 
 Section 
 
 III. 
 
 Euclid, . 
 
 • 
 
 Section 
 
 IV. 
 
 Mechanics, 
 
 • 
 
 Section 
 
 V. 
 
 Chemistry, 
 
 • 
 
 Section 
 
 VI. 
 
 Heat and Light, 
 
 • 
 
 Section VII. 
 
 Magnetism and Electricity 
 
WRINKLES’ IN ALGEBRA, ARITHMETIC, Ac. 
 
 SECTION I. 
 
 ALGEBRA. 
 
 1 . 
 
 (a) (a + b ) 2 
 
 (fi) (« - 6 ) 
 
 (y) (a + &) (ct-b) 
 
 a 2 4 - '2ab + 6 2 . 
 a 2 - lab + 6 2 . 
 
 a 2 - h 2 . 
 
 a 3 + 3a 2 6 + 3a6 2 + 6 3 . 
 a 3 - 3a 2 6 + 3a6 2 - 6 3 . 
 
 9 
 
 (a + by 
 (a - by 
 
 3. To square a trinomial such as (a + b - c), treat (a + b) 
 as one term, and then square as in (1) (3. 
 
 (a + b - c) 2 — {(a + b) - c} 2 = a 2 + ‘lab + b 2 - lac - 26c 4- c 2 . 
 
 4. To square a quantity with four terms as (a + 6 - c + d), 
 bracket so as to reduce it to form of (1) /?. 
 
 (a + b - c + cl) 2 = {(a + b) - (c - d)} 2 = 
 
 a 2 4 lab +b 2 — 1 (ac + bc- ad - bd) + c 2 - led + d 2 . 
 
 5. Let n be any whole number, then 
 
 / v a n -f b n is divisible by a + b when n is odd. 
 
 [CLjTl. -tfi 7 
 
 v 7 a + b if a - b never. 
 
 ( 0/71 ~ * s divisible by a + b when n is even. 
 
 'P' a n -b n ii a - b always. 
 
 6. When a n — b n is divided by a —b, all terms of 
 
 quotient will be + as --— = a 3 + a 2 b 4- a6 2 + 6 3 . 
 
 a-b 
 
 7. When a n - b n is divided by a + b, then the terms 
 
G 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 8. a° = 1. 
 
 9. Let x 2 +px + q = o be taken as type of quadratic equation, 
 then (a) a quadratic equation has two roots and only two. 
 
 ( b ) the roots of the equation are :— 
 
 i. 
 
 li. 
 
 -p.+ 
 
 9 
 
 \/(? - *) 
 -i -Ai- *) 
 
 iii. The sum of the roots = coefficient of second 
 
 term with sign changed. 
 
 iv. The product of the roots = the last term. 
 
 10. The equation to x 2 +px + q = 0 has its roots. 
 
 (a) real and different when p 2 is greater than 4^/. 
 
 (b) real and equal when p 2 is equal to 4 q. 
 
 (c) impossible and different when p 2 is less than 4 q. 
 
 11. Arith. Prog. 
 
 s = sum of terms, n = number of terms, a = first term, 
 d — common difference, l = last term. 
 
 Then (a) l = a + (n - 1 )d 
 
 08 )« = £ (a + l) 
 
 substituting value of l in (/?) s = ~ {2a+ (n - 1)^7} 
 
 A 
 
 12. Geom. Prog. 
 
 s = sum of terms, a = first term, r = constant factor, 
 n = number of terms. 
 
 Then i. n th term = ar n ~ x 
 
 a (r n - 1) a ( 1 - r n ) 
 
 li. s = —^- —a or —A- 
 
 r - l 1 - r 
 
 iii. To sum a series of terms to infinity 
 
 a 
 
 s — 
 
 1 - r 
 
 a + b 
 
 13. The Arith. mean between a and b is 
 
 A 
 
 ir Geom. ii J ab 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 7 
 
 SECTION II. 
 
 ARITHMETIC. 
 
 1. Abstract numbers are numbers in general, "and have 
 no reference to one thing more than another, as 9, 11. 
 
 2. Concrete numbers are applied to some particular 
 object, as 9 cows, 11 horses. 
 
 3. A prime number is one that can only be divided by 
 itself and unity without a remainder; and numbers are 
 prime to each other when no number but unity will divide 
 them without a remainder. 
 
 4. Composite numbers are such as are produced by the 
 multiplication of two or more prime numbers. 
 
 5. To multiply by 5, 25, 125, 625, 
 
 (a) To multiply by 5 add 1 cypher to the number and -=- by 2 
 
 (b) n 25 m 2 cyphers »» -f- u 4 
 
 (c) u 125 ii 3 n n -f- u 8 
 
 (d) n 625 n 4 a n t n 16 
 
 Similar rules can be found to multiply by 1*25, 62*5, 
 
 333J, &c. 
 
 6. To divide by 25, 125, <fec. 
 
 (а) 25 Multiply by 4 and cut off 2 figures as decimals. 
 
 (б) 125 „ 8 „ 3 
 
 (c) 625 „ 16 ,i 4 
 
 7. To multiply by 99, 999, &c. 
 
 (а) By 99 annex two cyphers to the multiplicand and 
 subtract the original multiplicand from it. 
 
 (б) By 999 annex three cyphers to the multiplicand and 
 subtract the original multiplicand from it. 
 
 8. To square numbers use formula ( a + b) 2 = a 2 4- 2 ah + b 2 
 thus: (907) 2 = (900 + 7) 2 = 810000 + 12600 + 49 = 822649. 
 
8 
 
 ‘ WEINKLES IN ALGEBEA, 
 
 9 If two numbers are to be multiplied together which 
 are respectively the sum and difference of two numbers, 
 it is advantageous to use formula (a + b) (a - b) = a 2 - b 2 
 thus: 287 x 313 = (300- 13) (300 + 13) = 90000 - 169 = 
 89831. 
 
 10. Any number is divisible by 2 without remainder if its 
 
 last figure is divisible by 2. 
 
 Any number is divisible by 4 without remainder if its 
 last 2 figures are divisible by 4. 
 
 Any number is divisible by 8 without remainder if its 
 last 3 figures are divisible by 8. 
 
 11. Any number is divisible by 5 without remainder if 
 
 its last figure is divisible by 5. 
 
 Any number is divisible by 25 without remainder if 
 its last 2 figures are divisible by 25. 
 
 Any number is divisible by 125 wdthout remainder if 
 its last 3 figures are divisible by 125. 
 
 12. Any number is divisible by 9 if the sum of its figures 
 
 be divisible by 9. 
 
 13. The tables of weights and measures should be known 
 thoroughly. The following are often useful:— 
 
 1 lb. Avoirdupois = 7000 grains Troy. 
 
 = 5760 
 
 = 1 fathom. 
 
 = 1 chain. 
 
 = 1 chain. 
 
 = 1 hand (used in measuring horses). 
 
 14. Rule for greatest common measure :— 
 
 (a) For two numbers:—Divide the greater by the less, 
 the divisor by the remainder, and so on until nothing 
 remains. The last divisor is the G.C.M. 
 
 (b) For more than two numbers :—Find the G.C.M. of 
 any two of them; then of this common measure and a third 
 number; then of this last found common measure and a 
 
 (a) 1 lb. Troy 
 
 ( b ) 6 feet 
 
 (c) 100 links 
 
 ( d) 22 yards 
 
 ( e ) 4 inches 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 9 
 
 fourth, &c. The last common measure found will be the 
 G.C.M. of all the numbers. 
 
 15. In least common multiple:—It is best in dividing the 
 numbers to begin with 2, and to repeat this as often as 
 possible, and then to divide by the other prime numbers in 
 a similar w^ay. 
 
 16. Decimals. 
 
 (1) To multiply a decimal by 10, 100, 1000, &c., we 
 remove the decimal point 1, 2, 3, places to the right. 
 
 (2) To divide a decimal by 10, 100, 1000, we remove the 
 decimal point 1. 2. 3. places to the left. 
 
 (3) Hule for division of decimals:—Divide as in integers, 
 annexing as many cyphers to the dividend as may be thought 
 necessary; then point off from the right of the quotient as 
 many decimal places as the dividend has more than the 
 divisor, making up any deficiency by prefixing cyphers. 
 
 (4) Recurring Decimals. 
 
 (a) If the denominator of a vulgar fraction in its lowest 
 terms be composed of 2 and 5, or of one of them, or of pow T ers 
 of 2 and 5, or one of them, then it can be brought out to a 
 finite or terminating decimal. If this be not the case then 
 the division of the numerator by the denominator will never 
 terminate, and a decimal of this kind is called recurring or 
 circulating. 
 
 ( b ) To convert a repeating decimal to a vulgar fraction. 
 
 Rule .—-For the numerator taken the given decimal, minus 
 
 the non-repeating part, and for the denominator put down 
 as many 9’s as there are figures in the repeating part, 
 followed by as many cyphers as there are figures in the non¬ 
 repeating part. 
 
 (c) Addition and subtraction of repeating decimals. 
 
 Rule .—Arrange as in terminating decimals, carry on the 
 
 repeating part to two or three more places than the number 
 required, taking care that the last figure retained in the 
 
10 ‘ WRINKLES ’ IN ALGEBRA, 
 
 answer be increased by one if the succeeding figure be five 
 or more. 
 
 (d) Multiplication and division of repeating decimals. 
 
 Rule .—Convert the decimals to vulgar fractions, multiply 
 or divide in the ordinary way, and then, if necessary, reduce 
 the result again to a decimal. 
 
 17. The square root of a number is a number which 
 multiplied by itself will give the given number. 
 
 18. Buie for Pointing. Point off the number into periods, 
 of two figures each, beginning at the right if a whole 
 number, and at the left if a decimal. If one figure remains 
 over at the left in integers it is considered a period, but if 
 one remains over at the right in decimals, a cypher is added 
 to complete the period. 
 
 19. It is of advantage to recognise the squares of the 
 following numbers at sight:— 
 
 13 2 = 
 
 169. 
 
 18 2 = 324. 
 
 II 
 
 <N 
 
 CO 
 
 CM 
 
 529. 
 
 14 2 = 
 
 196. 
 
 19 2 = 361. 
 
 24 2 = 
 
 576. 
 
 15 2 = 
 
 225. 
 
 20 2 = 400. 
 
 25 2 = 
 
 625. 
 
 16 2 = 
 
 256. 
 
 21 2 = 441. 
 
 26 2 = 
 
 676. 
 
 IT 2 = 
 
 289. 
 
 22 2 = 484. 
 
 27 2 = 
 
 729. 
 
 
 
 
 28 2 = 
 
 784. 
 
 
 
 
 29 2 = 
 
 841. 
 
 
 SECTION 
 
 I I I. 
 
 
 EUCLID. 
 
 A. General Explanations :— 
 
 1. A Proposition in Euclid is something required to be 
 
 done or to be proved. 
 
 2. A Problem proposes something to be done. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 11 
 
 3. A Theorem makes a statement, the truth of which 
 
 it is proposed to demonstrate. 
 
 4. A Postulate is a problem so simple that the 
 
 possibility of doing it is self-evident and requires 
 no proof. 
 
 5. An Axiom is a theorem, the truth of which is 
 
 admitted without proof. 
 
 6. A Corollary is an inference which rises immediately 
 
 from the discussion of the proposition to which 
 it is affixed. 
 
 Book I. 
 
 Prop. 1. Problem.—To describe an equilateral triangle 
 upon a given finite straight line. (Befs.: Def. 15; ax. 1; 
 posts. 1, 3.) 
 
 Prop. 2. Problem.—From a given point to draw a 
 straight line equal to a given straight line. (Refs.: Prop. 1; 
 posts. 1, 2, 3; ax. 1, 3; def. 15.) 
 
 Prop. 3. Problem.—From the greater of two given 
 straight lines to cut off a part equal to the less. (Refs.: 
 Prop. 2 ; post. 3; ax. 1 ; def. 15.) 
 
 Prop. 4. Theorem.—If two triangles have two sides of 
 the one equal to two sides of the other, each to each, and 
 have likewise the angles contained by those sides equal to 
 one another, they shall have their bases or third sides equal, 
 the two triangles shall be equal, and their other angles 
 shall be equal, each to each, viz., those to which the equal 
 sides are opposite. (Refs.: Axs. 8, 10.) 
 
 Prop. 5. Theorem.—The angles at the base of an isosceles 
 triangle are equal to one another; and if the equal sides be 
 produced, the angles upon the other side of the base shall 
 also be equal. (Refs.: Props. 3, 4; ax. 3.) 
 
 Corollary .— Hence every equilateral triangle is also equi¬ 
 angular. 
 
12 
 
 4 WPJNKLES ’ IN ALGEBKA, 
 
 Prop. 6. Theorem.—If two angles of a triangle be equa 
 to one another, the sides also which subtend, or are opposite 
 to the equal angles, shall be equal to one another. (Refs.: 
 Props. 3, 4.) 
 
 Corollary .—Hence every equiangular triangle is also equi¬ 
 lateral. 
 
 Prop. 7. Theorem.—Upon the same base and upon the 
 same side of it there cannot be two triangles which have 
 their sides, which are terminated in one extremity of the 
 base, equal to one another, and likewise those which are 
 terminated in the other extremity. (Refs.: Prop. 5 ; ax. 9.) 
 
 Prop. 8. Theorem.—If two triangles have two sides of 
 the one equal to two sides of the other, each to each, and 
 have likewise their bases equal, the angle which is contained 
 by the two sides of the one shall be equal to the angle con¬ 
 tained by the two sides equal to them of the other. (Refs.: 
 Prop. 7; ax. 8.) 
 
 Prop. 9. Problem.—To bisect a given rectilineal angle, 
 that is, to divide it into two equal parts. (Refs.: Props. 1, 
 3, 8.) 
 
 Prop. 10. Problem.—To bisect a given finite straight 
 line, that is, to divide it into two equal parts. (Refs.: 
 Props. 1, 4, 9.) 
 
 Prop. 11. Problem.—To draw a straight line at right 
 angles to a given straight line from a given point in the 
 same. (Refs.: Props. 1, 3, 8 ; ax. 1; def. 10.) 
 
 Corollary. —Two straight lines cannot have a common 
 segment. 
 
 Prop. 12. Problem.—To draw a straight line perpen¬ 
 dicular to a given straight line of unlimited length from a 
 given point without it. (Refs.: Props. 8, 10; post. 3; defs. 
 10, 15.) 
 
 Prop. 13. Theorem.—The angles which one straight line 
 makes with another upon one side of it are either two right 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 13 
 
 angles, or are together equal to two right angles. (Refs.: 
 Prop. 11; axs. 1, 2; clef. 10.) 
 
 Prop. 14. Theorem.—If, at a point in a straight line, two 
 other straight lines upon the opposite sides of it make the 
 adjacent angles equal together to two right angles, these 
 two straight lines shall be in one and the same straight line. 
 (Refs.: Prop. 13; axs. 1, 3.) 
 
 Prop. 15. Theorem.—If two straight lines cut one 
 another, the vertical or opposite angles shall be equal. 
 (Refs.: Prop. 13; axs. 1, 3.) 
 
 Corollary /.—If two straight lines cut one another, the 
 angles which they make at the point where they cut are 
 together equal to four right angles. 
 
 Corollary II. —And, consequently, that all the angles 
 made by any number of lines meeting in one point are 
 together equal to four right angles. 
 
 Prop. 16. Theorem.—If one side of a triangle be pro¬ 
 duced, the exterior angle shall. be greater than either of 
 the interior opposite angles. (Refs.: Props. 3, 4, 10, 15; 
 ax. 9.) 
 
 Prop. 17. Theorem.—Any two angles of a triangle are 
 together less than two right angles. (Refs.: Props. 13, 
 16; ax. 4.) 
 
 Prop. 18. Theorem.—The greater side of every triangle 
 is opposite the greater angle. (Refs.: Props. 3, 5, 16.) 
 
 Prop. 19. Theorem.—The greater angle of every triangle 
 is subtended by the greater side, or has the greater side 
 opposite to it. (Refs.: Props. 5, 18.) 
 
 Prop. 20. Theorem.—Any two sides of a triangle are 
 together greater than the third side. (Refs.: Props. 3, 5, 
 19; ax. 9.) 
 
 Prop. 21. Theorem.—If from the ends of the side of a 
 triangle there be drawn two straight lines to a point 
 within the triangle, these shall be less than the other 
 
14 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 two sides of the triangle, but shall contain a greater 
 angle. (Refs.: Props. 16, 20; ax. 4.) 
 
 Prop. 22. Problem.—To make a triangle of which the 
 sides shall be equal to three given straight lines, but 
 any two whatever of these lines must be greater than 
 the third. (Refs.: Prop. 3; post. 3; ax. 1; def. 15.) 
 
 Prop. 23. Problem. — At a given point in a given 
 straight line, to make a rectilineal angle equal to a given 
 rectilineal angle. (Refs.: Props. 8, 22.) 
 
 Prop. 24. Theorem.—If two triangles have two sides 
 of the one equal to two sides of the other, each to each, 
 but the angle contained by the two sides of one of them 
 greater than the angle contained by the two sides equal 
 to them of the other, the base of that which has the 
 greater angle shall be greater than the base of the other. 
 (Refs.: Props. 3, 4, 5, 19, 23 ; ax. 9.) 
 
 Prop. 25. Theorem.—If two triangles have two sides 
 of the one equal to two sides of the other, each to each, 
 but the base of the one greater than the base of the 
 other, the angle contained by the sides of that which has 
 the greater base shall be greater than the angle con¬ 
 tained by the sides equal to them of the other. (Refs.: 
 Props. 4, 24.) 
 
 Prop. 26. Theorem.—If two triangles have two angles 
 of the one equal to two angles of the other, each to each, 
 and one side equal to one side, viz., either the side 
 adjacent to the equal angles in each, or the side opposite 
 to them; then shall the other sides be equal, each to 
 each, and also the third angle of the one equal to the 
 third angle of the other. (Refs.: Props. 3, 4, 16; ax. 1.) 
 
 Prop. 27. Theorem.—If a straight line falling upon 
 two other straight lines makes the alternate angles equal 
 to one another, these two straight lines shall be parallel. 
 (Refs.: Prop. 16; def. 34.) 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 15 
 
 Prop. 28. Theorem.—If a straight line falling upon two 
 other straight lines make the exterior angle equal to the 
 interior and opposite, upon the same side of the line, or make 
 the interior angles upon the same side together equal to two 
 right angles, the two straight lines shall be parallel to one 
 another. (Refs.: Props. 13, 15, 27 ; axs. 1, 3.) 
 
 Prop. 29. Theorem.—If a straight line fall upon two 
 parallel straight lines, it makes the alternate angles equal 
 to one another, the exterior angle equal to the interior 
 and opposite upon the same side; and also the two in¬ 
 terior angles upon the same side equal to two right 
 angles. (Refs.: Props. 13, 15; axs. 1, 2, 4, 12.) 
 
 Prop. 30. Theorem.-—Straight lines which are parallel 
 to the same straight line are parallel to one another. 
 (Refs.: Props. 27, 29; ax. 1.) 
 
 Prop. 31. Problem.—To draw a straight line through 
 a given point, parallel to a given straight line. (Refs.: 
 Props. 23, 27.) 
 
 Prop. 32. Theorem.—If a side of any triangle be pro¬ 
 duced, the exterior angle is equal to the two interior and 
 opposite angles; and the three interior angles of every 
 triangle are equal to two right angles. (Refs.: Props. 13, 
 29, 31 ; axs. 1, 2.) 
 
 Corollary /.—All the interior angles of any rectilineal 
 figure, together with four right angles, are equal to 
 twice as many right angles as the figure 'has sides. 
 (Refs.: Prop. 15; cor. II.) 
 
 Corollary II .—All the exterior angles of any rectilineal 
 figure are together equal to four right angles. (Refs.: 
 Props. 13, 32 ; cor. I.; axs. 1, 3.) 
 
 Prop. 33. Theorem.—The straight lines which join the 
 extremities of two equal and parallel straight lines towards 
 the same parts, are also themselves equal and parallel. 
 (Refs.: Props. 4, 27, 29.) 
 
16 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 Prop. 34. Theorem.—The opposite sides and angles of 
 a parallelogram are equal to one another, and the diagonal 
 bisects it, that is, divides it into two equal parts. (Refs.: 
 Props. 26, 29; ax. 2.) 
 
 Prop. 35. Theorem. — Parallelograms upon the same 
 base and between the same parallels are equal to one 
 another. (Refs.: Props. 4, 29, 34; axs. 1, 2, 3, 6.) 
 
 Prop. 36. Theorem. — Parallelograms upon equal bases 
 and between the same parallels are equal to one another. 
 (Refs.: Props. 33, 34, 35; ax. 1.) 
 
 Prop. 37. Theorem.—Triangles upon the same base 
 and between the same parallels are equal to one another. 
 (Refs.: Props. 31, 34, 35 ; ax. 7.) 
 
 Prop. 38. Theorem.—Triangles upon equal bases and 
 between the same parallels are equal to one another. 
 (Refs.: Props. 31, 34, 36 ; ax. 7.) 
 
 Prop. 39. Theorem.—Equal triangles upon the same 
 base and on the same side of it are between the same 
 parallels. (Refs.: Props. 31, 37 ; ax. 1.) 
 
 Prop. 40. Theorem.—Equal triangles upon the same side 
 of equal bases that are in the same straight line are between 
 the same parallels. (Refs.: Props. 31, 38; ax. 1.) 
 
 Prop. 41. Theorem. — If a parallelogram and a triangle 
 be upon the same base and between the same parallels, 
 the parallelogram shall be double of the triangle. (Refs.: 
 Props. 34, 37 ; ax. 1.) 
 
 Prop. 42. Problem.—To describe a parallelogram that 
 shall be equal to a given triangle, and have one of its 
 angles equal to a given rectilineal angle. (Refs.: Props. 
 10, 23, 31, 38, 41 ; ax. 6; def. 35.) 
 
 Prop. 43. Theorem.—The complements of the parallelo¬ 
 grams which are about the diagonal of any parallelogram 
 are equal to one another. (Refs.: Prop. 34 ; axs. 2, 3.) 
 
 Prop. 44. Problem.—To a given straight line, to apply 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 17 
 
 a parallelogram which shall be equal to a given triangle, 
 and have one of its angles equal to a given rectilineal 
 angle. (Refs.: Props. 15, 29, 31, 42, 43; axs. 1, 9, 12.) 
 
 Prop. 45. Problem.—To describe a parallelogram equal 
 to a given rectilineal figure, and having an angle equal 
 to a given rectilineal angle. (Refs.: Props. 14, 29, 30, 
 42, 44; axs. 1, 2; def. 35.) 
 
 Prop. 46. Problem.—To describe a square upon a given 
 straight line. (Refs.: Props. 3, 11, 29, 31, 34; axs. 1, 
 3; def. 30.) 
 
 Prop. 47. Theorem.—In any right-angled triangle, the 
 square which is described upon the side opposite the right 
 angle, is equal to squares described upon the sides which 
 contain the right angle. (Refs.: Props. 4, 14, 31, 41, 46 ; 
 axs. 2, 6, 11 ; def. 30.) 
 
 Prop. 48. Theorem.—If the square described upon one 
 of the sides of a triangle be equal to the squares described 
 upon the other two sides of it, the angle contained by 
 these two sides is a right angle. (Refs.: Props. 3, 8, 
 11, 47 ; axs. 1, 2.) 
 
 Book II. 
 
 Prop. 1. Theorem.—If there be two straight lines, one 
 of which is divided into any number of parts, the rectangle 
 contained by the two straight lines is equal to rectangles 
 contained by the undivided line and the several parts of the 
 divided line. (Refs.: Props. I., 3, 11, 31, 34.) 
 
 Prop. 2. Theorem.—If a straight line be divided into any 
 two parts, the rectangles contained by the whole and each 
 of the parts are together equal to the square on the whole 
 line. (Refs.: I., 46, 31.) 
 
 Prop. 3. Theorem.—If a straight line be divided into any 
 two parts, the rectangle contained by the whole and one of 
 the parts is equal to the rectangle contained by the two 
 
 B 
 
18 ‘ WRINKLES ’ IN ALGEBRA, 
 
 parts, together with the square on the aforesaid part. (Refs.: 
 
 1., 46, 31.) 
 
 Prop. 4. Theorem.—If a straight line be divided into any 
 two parts, the square on the whole line is equal to the 
 squares on the two parts, together with twice the rectangle 
 contained by the two parts. (Refs.: I., 46, 31, 29, 5, 6, 
 34, 43; axs. 1, 3; def. 30.) 
 
 Prop. 5. Theorem.—If a straight line be divided into two 
 equal parts and also into two unequal parts, the rectangle 
 contained by the unequal parts, together with the square on 
 the line between the points of section, is equal to the square 
 on half the line. (Refs. : I., 46, 31, 43, 36, 34 ; axs. 1, 2 ; 
 
 11., 4, cor.) 
 
 Prop. 6. Theorem. — If a straight line be bisected and 
 produced to any point, the rectangle contained by the whole 
 line thus produced and the part of it produced, together 
 with the square on half the line bisected, is equal to the 
 square on the straight line which is made up of the half and 
 the part produced. (Refs.: I., 46, 31, 36, 43, 34; axs. 1, 2 : 
 
 II., 4, cor.) 
 
 Prop. 7. Theorem.—If a straight line be divided into any 
 two parts, the squares on the whole line and on one of the 
 parts are equal to twice the rectangle contained by the 
 whole and that part, together witli the square on the other 
 part. (Refs. : I., 43, 34; II., 4, cor.) 
 
 Prop. 8. Theorem. — If a straight line be divided into any 
 two parts, four times the rectangle contained by the whole 
 line and one of the parts, together w ith the square on the 
 other part, is equal to the square on the straight line w hich 
 is made up of the whole and that part. (Refs.: I., 3, 34, 
 36, 43; ax. 1 ; post. 2 ; II., 4, cor.) 
 
 Prop. 9. Theorem. — If a straight line be divided into tw o 
 equal and also into tw r o unequal parts, the squares on the 
 two unequal parts are together double of the square on half 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 19 
 
 the line and of the square between the points of section. 
 (Refs.: I., 11, 3, 31, 5, 32, 29, 6, 34, 47.) 
 
 Prop. 10. Theorem.—If a straight line be bisected and 
 produced to any point, the square on the whole line thus 
 produced and the square on the part of it produced are 
 together double of the square on half the line bisected and 
 of the square on the line made up of the half and the part 
 produced. (Refs.: I., 11, 3, 31, 29, 12, 5, 15, 32, 6, 34, 
 47 ; ax. 12.) 
 
 Prop. 11. Problem.—To divide a given straight line into 
 two parts, so that the rectangle contained by the whole and 
 one of the parts may be equal to the square on the other 
 part. (Refs. : I., 46, 10, 3, 47 ; ax. 3 ; II., 6.) 
 
 Prop. 12. Theorem.—In obtuse angle triangles, if a 
 perpendicular be drawn from either of the acute angles to 
 the opposite side produced, the square on the side sub¬ 
 tending the obtuse angle is greater than the squares on the 
 sides containing the obtuse angle by twice the rectangle 
 contained by the side on which, when produced, the per¬ 
 pendicular falls, and the straight line intercepted without 
 the triangle, between the perpendicular and the obtuse 
 angle. (Refs. : I., 47; ax. 2 ; II., 4.) 
 
 Prop. 13. Theorem.—In every triangle, the square on 
 the side subtending an acute angle is less than the squares 
 on the sides containing that angle by twice the rectangle 
 contained by either of these sides, and the straight line 
 intercepted between the perpendicular let fall on it from 
 the opposite angle and the acute angle. (Refs.: I., 47, 16 ; 
 ax. 2; II., 7, 12, 3.) 
 
 Prop. 14. Problem.—To describe a square that shall be 
 equal to a given rectilineal figure. (Refs.: I., 45, 3, 47, 10; 
 ax. 3; II., 5). 
 
20 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 Book III. 
 
 Prop. 1. Problem.—To find the centre of a given circle 
 (Kefs. : I., 10, 11, 8; ax. 11 ; defs. 10, 15.) 
 
 Corollary .—From this it is manifest, that if in a circle a 
 straight line bisect another at right angles, the centre of 
 the circle is in the straight line which bisects the other. 
 
 Prop. 2. Theorem.—If any two points be taken in the 
 circumference of a circle, the straight line which joins them 
 shall fall within the circle. (Refs.: I., 5, 16, 19; def. 15; 
 III., 1.) 
 
 Prop. 3. Theorem.—If a straight line drawn through the 
 centre of a circle bisect a straight line in it which does not 
 pass through the centre, it shall cut it at right angles ; and 
 if it cut it at right angles, it shall bisect it. (Refs.: I., 5, 
 8, 26; defs. 10,15; III, 1.) 
 
 Prop. 4. Theorem. — If in a circle two straight lines cut 
 one another, which do not both pass through the centre, 
 they do not bisect one another. (Refs.: Ax. 11 ; III, 1, 3.) 
 
 Prop. 5. Theorem. — If two circles cut one another, they 
 shall not have the same centre. (Refs.: Ax. 1 ; def. 15.) 
 
 Prop. 6. Theorem. — If two circles touch one another 
 internally, they shall not have the same centre. (Refs.: 
 Ax. 1 ; def. 15.) 
 
 Prop. 7. Theorem.—If any point be taken in the 
 diameter of a circle which is not the centre of all the 
 straight lines which can be drawn from this point to the 
 circumference, the greatest is that in which the centre is, 
 and the other part of the diameter is the least; and, of any 
 others, that which is nearer to the straight line which passes 
 through the centre is always greater than one more remote ; 
 and from the same point there can be drawn to the circum¬ 
 ference two straight lines, and only two, which are equal to 
 one another, one on each side of the shortest line. (Refs.: 
 I, 20, 24, 23, 4; ax. 1 ; def. 15.) 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 21 
 
 Prop. 8. Theorem.—If any point be taken without a 
 circle, and straight lines be drawn from it to the circum¬ 
 ference, one of which passes through the centre; of those 
 which fall on the concave circumference, the greatest is 
 that which passes through the centre; and of the rest, that 
 which is nearer to the one passing through the centre is 
 always greater than one more remote; but of those which 
 fall on the convex circumference, the least is that between 
 the point without the circle and the diameter; and of the rest, 
 that which is nearer to the least is always less than one 
 more remote; and from the same point there can be drawn 
 to the circumference two straight lines, and only two, which 
 are equal to one another, one on each side of the shortest 
 line. (Refs.: I., 20, 24, 23, 4, 21 ; ax. 1 ; def. 15 ; III., 1.) 
 
 Prop. 9. Theorem.—If a point be taken within a circle, 
 from which there fall more than two equal straight lines to the 
 circumference, that point is the centre of the circle. 
 (Refs.: III., 7.) 
 
 Prop. 10. Theorem.—One circumference of a circle can¬ 
 not cut another at more than two points. (Refs.: I., def. 
 15; III., 1, 9, 5.) 
 
 Prop. 11. Theorem.—If two circles touch one another 
 internally, the straight line which joins their centres, being 
 produced, shall pass through the point of contact. (Refs.: 
 I., 20; def. 15.) 
 
 Prop. 12. Theorem.—If two circles touch one another 
 externally, the straight line which joins their centres shall 
 pass through the point of contact. (Refs.: I., 20; ax. 2 ; 
 def. 15.) 
 
 Prop. 13. Theorem.—One circle cannot touch another at 
 more points than one, whether it touches it on the inside or 
 outside. (Refs.: I., 10, 11; III., 2, 1, 11.) 
 
 Prop. 14. Theorem.—Equal straight lines in a circle are 
 equally distant from the centre, and those which are 
 
22 
 
 ‘ WPJNKLES ’ IN ALGEBRA, 
 
 equally distant from the centre are equal to one another. 
 (Refs.: I., 12, 47; axs. 1, 3, 6, 7; def. 15; III., 1, 3; def. 4.) 
 
 Prop. 15. Theorem.—The diameter is the greatest 
 straight line in a circle; and, of all others, that which is 
 nearer to the centre is always greater than one more 
 remote ; and the greater is nearer to the centre than the 
 less. (Refs.: I., 12, 20; ax. 2; def. 15; III., def. 5.) 
 
 Prop. 16. Theorem.—The straight line drawn at right 
 angles to the diameter of a circle from the extremity of it, 
 falls without the circle; and no straight line can be drawn 
 from the extremity, between that straight line and the 
 circumference, so as not to cut the circle. (Refs.: I., 5, 17, 
 12, 19; def. 15.) 
 
 Corollary .—The straight line drawm at right angles to 
 the diameter of a circle from the extremity of it touches 
 the circle. (Refs.: IIP, def. 2; IIP, 2.) 
 
 Prop. 17. Problem.—To draw a straight line from a 
 given point, either without or in the circumference, which 
 shall touch a given circle. (Refs.: I., 11, 4; ax. 1; def. 15; 
 IIP, 1 and 16 cor.) 
 
 Prop. 18. Theorem.—If a straight line touch a circle, the 
 straight line drawn from the centre to the point of contact 
 shall be perpendicular to the line touching the circle. (Refs.: 
 I., 17, 19; def. 15.) 
 
 Prop. 19. Theorem.—If a straight line touch a circle, and 
 from the point of contact a straight line be drawn at right 
 angles to the touching line, the centre of the circle shall be 
 in that line. (Refs.: Ax. 11; IIP, 18.) 
 
 Prop. 20. Theorem.—The angle at the centre of a circle 
 is double of the angle at the circumference on the same base, 
 that is on the same arc. (Refs.: I., 5, 32.) 
 
 Prop. 21. Theorem.—The angles in the same segment o_ 
 a circle are equal to one another. (Refs.: Axs. 2, 7; IIP, 1, 20.) 
 
 Prop. 22. Theorem. — The opposite angles of any quadri- 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 23 
 
 lateral figure inscribed in a circle are together equal to two 
 right angles. (Refs.: I., 32 ; ax. 2 ; III., 21.) 
 
 Prop. 23. Theorem.—On the same straight line and on 
 the same side of it there cannot be two similar segments of 
 circles not coinciding with one another. (Refs.: L, 16, III., 
 10; def. 11. 
 
 Prop. 24. Theorem.—Similar segments of circles on equal 
 straight lines are equal to one another. (Refs.: III., 23.) 
 
 Prop. 25. Problem.—A segment of a circle being given, 
 to describe the circle of vdiich it is a segment. (Refs. : I., 10, 
 
 II, 6, 23, 4; ax. 1; III., 9.) 
 
 Prop. 26. Theorem.—In equal circles, equal angles stand 
 on equal arcs, whether they be at the centres or circumfer¬ 
 ences. (Refs.: I., 4; ax. 3; III., 24; defs. 1 and 11.) 
 
 Prop. 27. Theorem.—In equal circles, the angles which 
 stand on equal arcs are equal to one another, v 7 hether they 
 be at the centres or circumferences. (Refs.: I., 23; axs. 7, 1; 
 
 III. , 20, 26.) 
 
 Prop. 28. Theorem.—In equal circles equal straight lines 
 cut off equal arcs, the greater equal to the greater, and the 
 less equal to the less. (Refs.: I., 8; ax. 3; III., 1, 26; def. 1.) 
 
 Prop. 29. Theorem.—In equal circles, equal arcs are sub- 
 tendedby equal straight lines. (Refs.: I., 4; III., 1, 27; def. 1.) 
 
 Prop. 30. Prob.—To bisect a given arc, that is to divide it 
 into two equal parts. (Refs.: I., 10, 11,4; III., 1, cor. and 28.) 
 
 Prop. 31. Theorem.—In a circle the angle in a semicircle 
 is a right angle; but the angle in a segment greater than a 
 semicircle is less than a right* angle; and the angle in a 
 segment less than a semicircle is greater than a right angle. 
 (Refs.: I, 5, 32, 17; ax. 2; defs. 15, 10; III., 22.) 
 
 Corollary. —If one angle of a triangle equal the other tw r o, 
 it is a right angle. (Refs.: I., 32, and def. 10.) 
 
 Prop. 32. Theorem.—If a straight line touch a circle and 
 from the point of contact a straight line be drawn, cutting 
 
24 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 the circle, the angles which this line makes with the line 
 touching the circle shall be equal to the angles in the 
 alternate segments of the circle. (Refs.: I., 11, 32, 13; ax. 3; 
 III., 19, 31, 22.) 
 
 Prop. 33. Problem.—On a given straight line to describe 
 a segment of a circle containing an angle equal to a given 
 rectilineal angle. (Refs.: I., 10, 11, 4, 23; ax. 1; def. 10; 
 III., 31, 16 cor. and 32.) 
 
 Prop. 34. Problem.—From a given circle to cut off a 
 segment containing an angle equal to a given rectilineal 
 angle. (Refs.: I., 23; ax. 1; IIP, 17. 32.) 
 
 Prop. 35. Theorem.—If two straight lines cut one 
 another within a circle, the rectangle contained by the seg¬ 
 ments of one of them shall be equal to the rectangle contained 
 by the segments of the other. (Refs.: I., 12, 47; axs. 1, 2, 
 3; II., 5; III., 1, 3.) 
 
 Prop. 36. Theorem.—If from any point without a circle 
 two straight lines be drawn, one of which cuts the circle and 
 the other touches it, the rectangle contained by the whole 
 line which cuts the circle and the part of it without the circle 
 shall be equal to the square on the line which touches it. 
 (Refs.; I., 12, 47; axs. 2, 3; II., 6; III., 18, 1, 3.) 
 
 Prop. 37. Theorem.—If from any point without a circle 
 there be drawm two straight lines, one of which cuts the 
 circle and the other meets it, and if the rectangle contained 
 by the whole line which cuts the circle, and the part of it 
 without the circle be equal to the square on the line which 
 meets the circle, the line whtch meets the circle shall touch 
 it. (Refs.: I., 8; ax. 1; def. 15; III., 17, 1, 18, 36, and 16 cor.) 
 
 Book IV. 
 
 Prop. 1. Problem.—In a given circle, to place a straight 
 line equal to a given straight line, which is not greater than 
 the diameter of the circle. (Refs.: I., 3; ax. 1; def. 15 ) 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 25 
 
 Prop. 2. Problem.—In a given circle, to inscribe a triangle 
 equiangular to a given triangle. (Kefs.: I., 23, 32; ax. 11, 3, 
 1; III., 7, 32.) 
 
 Prop. 3. Problem.—About a given circle, to describe a 
 triangle equiangular to a given triangle. (Refs.: I., 23, 13, 
 32; ax. 3, 11; III., 1, 17, 18.) 
 
 Prop. 4. Problem.—To inscribe a circle in a given 
 triangle. (Refs.: I., 9, 12, 26; axs. 11, 1; IIP, 16 cor.) 
 
 Prop. 5. Problem.—To describe a circle about a given 
 triangle. (Refs.: I., 10, 11, 4; ax. 1; III., 31 cor.) 
 
 Prop. 6. Problem.—To inscribe a square in a given circle. 
 (Refs.: I., 11, 4; IIP, 1, 31.) 
 
 Prop. 7. Problem.—To describe a square about a given 
 circle. (Refs.: I., 11, 28, 34; IIP, 1, 17, 18.) 
 
 Prop. 8. Problem.—To inscribe a circle in a given square. 
 (Refs.: I., 10, 31, 34; ax. 7; def. 30; IIP, 16 cor.) 
 
 Prop. 9. Problem.—To describe a circle about a given 
 square. (Refs.: I., 8, 6; ax. 7.) 
 
 Prop. 10. Problem.—To describe an isosceles triangle, 
 Paving each of the angles at the base double of the third 
 angle. (Refs.: I., 32, 5, 6; ax. 1,2, 6; II., 11; IIP, 32, 37; 
 IV., 1, 5.) 
 
 Prop. 11. Problem.—To inscribe an equilateral and 
 equiangular pentagon in a given circle. (Refs.: I., 9; ax. 2; 
 IIP, 26, 27, 29; IV., 2, 10.) 
 
 Prop. 12. Problem.—To describe an equilateral and 
 equiangular pentagon about a given circle. (Refs.: I., 47, 8, 
 4, 26; axs. 1, 7, 36; IIP, 17, 18, 27.) 
 
 Prop. 13. Problem.—To inscribe a circle in a given 
 
 equilateral and equiangular pentagon. (Refs. I., 9, 4, 12, 26; 
 IIP, 16.) 
 
 Prop. 14. Problem.—To describe a circle about a given 
 equilateral and equiangular pentagon. (Refs.: I., 9, 6; 
 ax. 7.) 
 
26 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 Prop. 15. Problem.—To inscribe an equilateral and 
 equiangular hexagon in a given circle. (Refs.: I., 5, cor. 32, 
 13, 15; ax. 1; III., 1, 26, 29, 27.) 
 
 Prop. 16. Problem.—To inscribe an equilateral and 
 equiangular quindecagon in a given circle. (Refs.: III., 30; 
 
 IV, 2, 11, 1.) 
 
 SECTION IV. 
 
 MECHANICS. 
 
 A. Dynamics— 
 
 Def. of Velocity. —The velocity of a point in motion at 
 any moment is the degree of quickness of the motion of the 
 point at that moment. 
 
 Laws of Motion — 
 
 i. Every body continues in a state of rest or of uniform 
 motion in a straight line, except in so far as it may be com¬ 
 pelled to change that state by force acting upon it. 
 
 ii. Change of motion is proportional to the impressed 
 force, and takes place in the direction of the straight line in 
 which the force acts. 
 
 iii. Action and re-action are equal and opposite; that is, 
 to every action there is a corresponding re-action equal in 
 magnitude but opposite in direction. 
 
 Acceleration. —Acceleration is used as an abbreviation for 
 4 velocity added in a moment of time.’ 
 
 Formulae. — s = space ; v = velocity ; t = time in seconds. 
 
 Then (a) s = vt. 
 
 (b) Space described in the n th second under 
 acceleration 4 /.’ 
 
 * = |/(2 W -1). 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 27 
 
 (c) Space described in n seconds under accelera¬ 
 tion 
 8 = \ft 2 . 
 
 (</) = 2 / 5 . 
 
 If ‘g’ (32 feet) be substituted for in the formulae 
 above, they will then answer * for bodies moving under the 
 action of gravity. 
 
 (a) s = y(2n-l); ( b') s = \g$; (c) v 2 = 2gs. 
 
 Formulae for. motion of bodies in a straight line under 
 the influence of a uniform force, with a given initial 
 velocity. 
 
 The formulae will correspond to those given above 
 a, 6, c, d. 
 
 u — the given initial velocity. 
 
 (a) v — u± ft. 
 
 (b) s — u±\f(^n - 1). 
 
 (c) s = ut± J ft 2 . 
 
 (d) v 2 = u 2 ± 2/5. 
 
 N.B .—(a) The original velocity u is destroyed in time - r 
 
 u 2 
 
 (b ) and the space described in that time is 
 
 Most important use of this note is in problems respecting 
 gravity. 
 
 Force. —Force is that which produces acceleration in mass. 
 
 Force = mf (m = mass, and/= acceleration). 
 
 Mass .—The mass of a body or particle is the quantity of 
 matter which it contains. 
 
 Momentum .—Momentum is the product of mass moved 
 x by velocity. 
 
 Work. —Work is the production of motion against re¬ 
 sistance. 
 
 Energy .—Energy is the capacity for doing work. Energy 
 may be Potential or Kinetic. 
 
‘ WRINKLES ’ IN ALGEBRA, 
 
 (a) Potential Energy is the capacity for doing work 
 possessed by mass in virtue of its position. 
 
 ( b ) Kinetic Energy is the capacity for doing work 
 possessed by mass in virtue of its mass velocity. 
 
 Conservation of Energy. —Energy is indestructible ; it can 
 change its form, but its quantity remains unchanged. 
 
 Parallelogram of Velocities .—If a body have communicated 
 to it simultaneously two velocities which are represented in 
 magnitude and direction by two straight lines drawn from 
 a point, then the resultant velocity will be represented in 
 magnitude and direction by the diagonal drawn from that 
 point of which the two straight lines are adjacent sides of 
 the parallelogram. 
 
 Parallelogram of Accelerations .—This may be deduced from 
 the Parallelogram of Velocities by reading ‘acceleration’ 
 for ‘ velocity.’ 
 
 Kf 
 
 Parallelogram of Forces. —If two forces acting on a particle 
 be represented in magnitude and direction by two straight 
 lines drawn from a point, and a parallelogram be con¬ 
 structed having these straight lines as adjacent sides, then 
 the resultant of the two forces is represented in magnitude 
 and direction by that diagonal of the parallelogram which 
 passes through the point. 
 
 Triangle of Forces. —If three forces acting on a particle be 
 represented in magnitude and direction by the sides of a tri¬ 
 angle taken in order, they will keep the particle in equilibrium. 
 
 Moments. —The moment of a force about a point is the 
 name given to power which that force has to turn any body 
 round that point. 
 
 In estimating the power of force to turn a body round a 
 fixed point we must consider— 
 
 1. The magnitude of the force. 
 
 2. The perpendicular distance of line of action of the 
 
 force from the fixed point. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 29 
 
 Propositions on Moments .— 
 
 I. The moment of a force about a point in its own line 
 of action is zero. 
 
 II. The algebraic sum of the moments of two parallel 
 forces about any point in their plane equals the moment of 
 their resultant about that point. 
 
 III. The algebraic sum of the moments of two forces 
 meeting in a point and acting in one plane about any 
 point in the plane, equals the moment of their resultant 
 about that point. 
 
 IV. The algebraic sum of the moments of two forces 
 acting in one plane and meeting in a point about any 
 point in the line of action of the resultant is zero. 
 
 B. Statics.— 
 
 Centre of Gravity. —The point at which the weight of a 
 body or system of bodies may always be supposed to act is 
 called its ‘ mass-centre 5 or * centre of gravity/ 
 
 Centre of Gravity of Parallelogram. —1. The centre of 
 gravity of a o is the intersection of the straight lines 
 which join the middle points of opposite sides. 
 
 Centre of Gravity of Triangle. —2. The centre of gravity 
 of a triangle is in the line joining middle point of base to 
 opposite angle, and J of its length from base. 
 
 The centre of gravity of a triangle coincides with centre 
 of gravity of 3 equal heavy particles placed at angular 
 points of the triangle. 
 
 Centre of Gravity of Cone. —3. The centre of gravity of 
 cone is in the line joining centre of gravity of base with the 
 vertex, and | of its length up. 
 
 Centre of Gravity of Sphere. —4. The centre of gravity of 
 a sphere is the same as its geometrical centre. 
 
 Equilibrium. — 
 
 (a) A body is said to be in unstable equilibrium when, on the 
 slightest disturbance, it moves away from its former position. 
 
30 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 (b) A body is said to be in stable equilibrium if it returns 
 to its former position. 
 
 (c) If a body remains in the new position which the 
 displacement has given it, the position is said to be neutral 
 
 A body placed with its base upon a plane surface will 
 stand or fall according as the vertical line through its centre 
 of gravity falls within or without the base. 
 
 C. Hydrostatics.— 
 
 1. Pascal?s Law — 
 
 Pressure exerted upon a mass of liquid is— 
 
 (a) Transmitted undiminished in all directions. 
 
 ( b ) Acts with the same force on all equal surfaces. 
 
 (c) And in a direction at right angles to those surfaces. 
 
 2. Pressure in liquids under gravity — 
 
 (a) The pressure in each layer is proportional to depth. 
 
 (b) With different liquids and same depth, pressure is 
 
 proportional to density of liquid. 
 
 ( c ) Pressure is the same at all points of the same 
 
 horizontal laver. 
 
 The upward pressure of liquids is governed by the same 
 laws as the downward pressure. 
 
 Equilibrium of a ; Liquid. —In order that a liquid may 
 remain at rest 
 
 (a) Its surface must be everywhere perpendicular to 
 
 the resultant of the forces which act on the 
 molecules of the liquid. 
 
 ( b) Every molecule of the mass of the liquid must be 
 
 subject in every direction to equal and contrary 
 pressures. 
 
 Archimedes' Principle .—A body immersed in a liquid loses 
 a part of its weight equal to the weight of displaced liquid. 
 
 Conditions of Equilibrium in Floating Bodies .— 
 
 I. The floating body must displace a volume of liquid 
 whose weight equals that of the body. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 31 
 
 II. The centre of gravity of the floating body must be in 
 the same vertical line as that of the fluid displaced. 
 
 III. The equilibrium of a floating body is stable or un¬ 
 stable as the metacentre is above or below the centre of 
 gravity. 
 
 Specific Gravity .—The Specific Gravity of a body is the 
 number wdiich expresses the relation of the weight of a 
 given volume of this body to the weight of the same volume 
 of distilled water at 4° C. 
 
 There are three methods of determining the specific 
 gravity of solids and liquids. 
 
 (a) Method of Hydrostatic Balance. 
 
 ( b ) n Hydrometer. 
 
 (c) i» Specific Gravity Bottle. 
 
 Formulae. — (a) To compare the Specific Gravities of a solid 
 and a fluid by means of Hydrostatic Balance. 
 
 Case I. When the solid is of greater Specific Gravity than 
 the fluid— 
 
 Let v = Yol. of solid. 
 m s — Specific Gravity of solid 
 u .s' = it fluid 
 
 H to — weight of solid in air 
 it w — it fluid 
 
 Case II. When the solid is of less Specific Gravity than 
 the fluid— 
 
 Attach a ‘sinker’ to the solid, which will make the 
 solid sink with it in the fluid— 
 
 Let w — weielit of the two bodies in air. 
 
 i n 
 
 II 
 
 IJ 
 
 II 
 
 II 
 
 X = 
 
 II 
 
 II 
 
 y = 
 
 II 
 
 II 
 
 /** _ 
 
 Xj - 
 
 II 
 
 solid in air. 
 two bodies in fluid, 
 sinker in air. 
 ii fluid. 
 
 // 
 
 5 IV 
 
 s' w - % — y + z 
 
 then 
 
32 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 Nicholson's Hydrometer. — (b) I. To compare Specific 
 Gravities of a solid and a fluid. 
 
 Let w = weight of substance which causes the hydrometer 
 to sink in fluid to the mark on the steel wire when placed 
 in upper dish. 
 
 Let x = weight which must be added to the solid whose 
 
 o 
 
 Specific Gravity is to be found to make the hydrometer 
 sink in fluid to the mark on the steel wire. 
 
 Let y = weight of substance which must be put in upper 
 dish to make the hydrometer sink in fluid to mark on the 
 steel wire when the solid is placed in the lower dish, 
 
 then Specific Gravity of solid w - x 
 Specific Gravity of fluid ~ y - x 
 
 II. To compare the Specific Gravities of two fluids by 
 weighing the same solid in each. 
 
 Let s and s = Specific Gravities of the fluids, 
 n w and w = weights of the solid when immersed in 
 
 O 
 
 respective fluids, 
 ii W= weight of solid in air— 
 
 then 
 
 W - iv 
 
 s' ~ W - w 
 
 D. Gases. Boyle and Mariotte's Law— 
 
 The temperature remaining the same, the volume of a 
 given quantity of gas is inversely as the pressure which it 
 supports. 
 
 But this law has now been proved to be only approxi¬ 
 mately true. 
 
 Charles Law .—All gases expand of their volume at 
 0° 0 for every increase in temperature of 1° C. 
 
 This fraction is called the ‘ coefficient of expansion of 
 gases. ’ 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 33 
 
 Graham's Law .—The rates of the diffusion of gases are 
 inversely proportional to their densities. 
 
 Atmospheric Pressure .—The pressure of the atmosphere 
 = 14*7 lbs. on a square inch—roughly speaking 15 lbs. 
 
 Suction Pump .—When water fills the pipe and barrel as 
 far as the spout, the effort necessary to raise piston = the 
 weight of a column of water, the base of which is the piston 
 and the height the vertical distance of the spout from the 
 level of water in reservoir, that is, the height to which the 
 water is raised. 
 
 SECTION V. 
 
 CHEMISTRY. 
 
 Chemical Combination— 
 
 1. Chemical Combination consists in the union of the 
 small indivisible particles of matter called ‘atoms/ These 
 atoms are all supposed to be of the same size with the 
 exceptions— 
 
 Phosphorus and Arsenic, which are half the usual size ; 
 Zinc, Cadmium, and Mercury, which are double 
 the usual size. 
 
 2. Avogadro’s Law— 
 
 The densities of all the elements known in a gaseous 
 state are identical with their atomic weights. 
 
 The vapour densities of compound substances are nearly 
 always half their molecular weight. 
 
 c 
 
34 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 3. Laws of Chemical Combination— 
 
 A. Law of Constant Proportion. 
 
 The same body is invariably composed of the same 
 elements united in the same proportions. 
 
 B. Law of Multiple Proportion. 
 
 When an element unites with another in different pro¬ 
 portions, the higher proportions are invariably multiples 
 of the lowest. 
 
 C. Law of Reciprocal Proportion. 
 
 If two bodies, A and B, separately combine with a third 
 body C, the proportions of A and B which unite with C 
 are measures or multiples of the proportions in which A and 
 B combine together. 
 
 Def. of Acid. —An acid is a body containing Hydrogen, 
 which may be replaced by a metal (or group of elements) 
 when presented to it in the form of an oxide or hydrate. 
 
 Acids containing one atom of H are called Monobasic, 
 it ii two atoms n Dibasic, 
 
 ii ii three atoms i? Tribasic. 
 
 Def of Base. —A base is a confound, usually an oxide or 
 hydrate, containing a metal (or group of elements) capable 
 of replacing the H of an acid. Bases are generally metallic 
 oxides. 
 
 Def. of an Alkali. —An alkali is a base of a specially 
 active character, soluble in water, to which it imparts a 
 soapy taste and touch. Alkalies restore the blue colour to 
 reddened litmus. 
 
 Def. of Hydrates. —Hydrates are mostly compounds of 
 metallic oxides with water. 
 
 Def. of a Salt. —A salt is the product of the action of 
 an acid on a base. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 35 
 
 Modes of Chemical Action— 
 
 I. Direct union. 
 
 II. Chemical displacement. 
 
 III. Mutual exchange. 
 
 IV. Re-arrangement of particles. 
 
 V. Direct decomposition. 
 
 Composition of the Atmosphere— 
 
 (a) Oxygen 
 Nitrogen 
 
 20-61. 
 
 77*95. 
 
 *04. 
 
 1*4. 
 
 ) 
 
 = Traces. 
 ~~ | Traces. 
 
 by volume, 
 by weight. 
 
 Carbonic dioxide 
 Aqueous vapour 
 Nitric Acid 
 Ammonia 
 
 Carburetted Hydrogen 
 In towns and manu- j Sulphuretted Hydrogen 
 facturing districts. { Sulphur dioxide 
 
 (6) The average composition of the atmosphere when 
 freed from C0 2 and Aqueous vapour is— 
 
 Nitrogen, . 79*19 
 
 Oxygen, . 20*81 
 
 Nitrogen, . 76*99 
 
 Oxygen, . 23*01 
 
 Notes. — (a) In Chemical calculations the Metric System 
 is used. 
 
 The Greek prefixes deca, hecto, kilo represent 10, 100, 
 and 1000 respectively. 
 
 The Latin deci, centi, milli represent a tenth, a 
 
 hundredth, and a thousandth respectively. 
 
 ( b ) 1 metre = 39*37 inches. 
 
 (c) 1 litre = 61*027 cubic inches or 1*7607 pints. 
 
 A litre is the volume occupied by a kilogram of water at 
 4° C., and it is equal to a cubic decimetre. 
 
 . \ 1 gram = 1 cubic centimetre. 
 
 (d) 1 gram = 15*4323 grains. 
 
36 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 (e) 1 cubic foot of water at 4° C. weighs 1000 ozs. 
 
 (f) 1 litre of H at 0° C. and 760 mm. pressure = crith — 
 *0896 gram. 
 
 (g) Prof. Williamson’s standard— 
 
 11*2 litres of H = 1 gram. 
 m m 0 = 16 grams. 
 
 &c. &c. &c. 
 
 SECTION VI. 
 
 HEAT AND LIGHT. 
 
 A. Heat— 
 
 I. Theories as to its Nature— 
 
 (a) The Emission or Material Theory supposes— 
 
 1. Heat consists of an imponderable substance sur¬ 
 
 rounding the molecules of bodies. 
 
 2. These heat atmospheres are self-repellant, and 
 
 therefore heat acts in opposition to the force ol 
 cohesion. 
 
 ( b ) The Dynamical Theory is the one universally held, 
 
 and supposes— 
 
 1. Heat to be caused by the rapid vibratory motion 
 
 of its particles. 
 
 2. The hottest bodies are those in which these 
 
 vibrations have the greatest velocity and ampli¬ 
 tude. 
 
 3. The molecules of warm bodies are capable of 
 
 communicating a vibratory motion to the im¬ 
 ponderable elastic ether (which is assumed to 
 pervade all matter and all space), and thus 
 neighbouring bodies become heated. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 37 
 
 II. Expansion of Solids, Liquids, and Gases— 
 
 Coefficient of Expansion. — (a) The Coefficient of Ex¬ 
 pansion is that fraction of a body’s length which it expands 
 on being heated 1° C. 
 
 The coefficient of cubical expansion is approximately 
 three times the linear expansion. 
 
 Boiling Point of Liquid. —( b ) The boiling point of a 
 liquid is that point of temperature at which the tension of 
 its vapour equals the pressure it supports. 
 
 The boiling point is lowered 1° C. for every 1060 feet of 
 elevation. 
 
 Maximum Density of Water. —( c ) Water is at its maximum 
 density at 4° C. 
 
 Gay Lussads Laics — 
 
 (d) (1) All gases have the same coefficient of expansion 
 
 as air. 
 
 (2) This coefficient is the same whatever be the 
 
 pressure supported by the gas. (This coefficient 
 
 = i i or -A \ 
 
 30 00 2 73 /* 
 
 III. Thermometry— 
 
 Temperature — 
 
 (а) The temperature of a body is the greater or less 
 extent to which it tends to impart sensible heat to other 
 bodies. 
 
 Thermometers — 
 
 (б) (1) To convert Fahrenheit degrees to Centigrade 
 
 (F. - 32) | = C. 
 
 (2) To convert Centigrade degrees to Fahrenheit 
 
 (| C. + 32) = F. 
 
 (3) To convert Reaumur degrees to Fahrenheit 
 
 (f R. + 32) = F. 
 
 (4) To convert Fahrenheit degrees to Reaumur 
 
 (F.-32) * =R. 
 
38 
 
 £ WRINKLES ’ IN ALGEBRA, 
 
 IV. Calorimetry and Specific Heat— 
 
 Thermal Unit. —(a) The thermal unit is the quantity of 
 heat required to raise 1 lb. of water 1° C. at the standard 
 temperature. 
 
 Specific Heat. — (b) The Specific Heat of a body is the 
 relation between the quantity of heat required to raise 
 the body and that required to raise an equal weight of 
 water through 1° C. 
 
 Y. Solidification and Liquefaction— 
 
 Laws of Solidification — 
 
 (a) 1. Every body under the same pressure solidifies 
 
 at a fixed temperature, which is the same as 
 that of fusion. 
 
 2. From the commencement to the end of solidi¬ 
 fication the temperature of a liquid remains 
 the same. 
 
 Laics of Liquefaction or Fusion — 
 
 ( b ) 1. Every body under the same pressure begins to 
 
 fuse at a fixed temperature. 
 
 2. From the commencement to the end of lique¬ 
 faction the temperature of the substance 
 remains the same. 
 
 YI. Vaporization, Condensation, and Ebullition— 
 
 (a) Vaporization is used to denote the change which 
 
 takes place when a liquid passes into the 
 gaseous state. It includes Evaporation and 
 Ebullition. 
 
 ( b ) The condensation of vapours may be due to cooling, 
 
 compression, or chemical affinity. 
 
 (c) Laws of Ebullition — 
 
 1. The temperature of Ebullition, or the boiling 
 point, increases with the pressure. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 39 
 
 2. All liquids begin to boil at a temperature which 
 
 for each liquid is constant under a constant 
 pressure. 
 
 3. As soon as ebullition begins, the temperature 
 
 of the liquid remains stationary. 
 
 (d) Laws of Evaporation — 
 
 1. Evaporation takes place more rapidly the 
 
 greater the surface exposed. 
 
 2. It increases with the increase of temperature. 
 
 3. It decreases as the air becomes more and more * 
 
 saturated with the same vapour. 
 
 VII. Latent Heat— 
 
 Latent Heat is the quantity of heat which disappears, and 
 is not shown on the thermometer when the molecular con¬ 
 stitution of a body is being changed. 
 
 (а) Latent Heat of Fusion of Water is 80 thermal units. 
 
 (б) ii n Evaporation n 536 n u 
 
 VIII. Conduction and Convection— 
 
 (a) Heat is transmitted by Conduction when each atom 
 
 takes up the motion of its neighbour and sends 
 it on to others. 
 
 (b) Heat is transmitted by Convection when it is pro¬ 
 
 pagated in a liquid or gas by the motion of 
 its heated particles. 
 
 B. Light—. 
 
 Theories— 
 
 I. Emission Theory —supported by Newton— 
 
 (a) Light consists of an imponderable substance, con¬ 
 
 sisting of corpuscles of matter. 
 
 ( b ) These corpuscles are emitted from luminous bodies 
 
 with enormous velocity, and by their impact on 
 the retina of the eye produce the sensation of vision. 
 
40 
 
 ‘ WRINKLES ’ IN ALGEBRA, 
 
 II. The Undulatory Theory —now generally accepted— 
 {a) There exists in space an exceedingly rarefied but 
 very elastic substance called ‘ ether/ which also 
 surrounds the molecules of matter. 
 
 ( b ) The particles of luminous bodies are in a state of 
 
 extremely rapid vibration. 
 
 (c) These vibrations are generated in the ether as 
 
 undulations, and proceed with great velocity in 
 concentric spheres. 
 
 (d) By the impact of these waves upon the retina, the 
 
 sensation of vision is caused. 
 
 Velocity of Light — 
 
 Boemer’s method gave 192,000 mis. per second. 
 Bradley’s n m 190,860 n n 
 
 Foucault’s m m 185,157 n n 
 
 Fizeau-Cornu’s n u 186,616 n n 
 
 Intensity of Light — Laws — 
 
 1. The intensity of illumination is inversely as the square 
 of its distance. 
 
 2. The intensity of illumination which is received 
 obliquely is proportional to the cosine of the angle which 
 the luminous rays make with the normal to the illuminated 
 surface. 
 
 Photometers. —A Photometer is an apparatus for measur¬ 
 ing the relative intensities of light. The principal are 
 Bumford’s, Bunsen’s, and Wheatstone’s. 
 
 Reflection at plane or spherical surfaces — 
 
 (a) Laws of the Beflection of Light— 
 
 1. The angle of reflection equals the angle of 
 
 incidence. 
 
 2. The incident and the reflected rays are both in 
 
 the same plane, and this is perpendicular to 
 the reflecting surface. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 41 
 
 (b) Relative velocity of mirror and image— 
 
 1. If a mirror be moved parallel to itself, either 
 
 from or towards an object, the image moves 
 twice as fast as the mirror. 
 
 2. If a mirror be made to rotate, the angle through 
 
 which the image moves is twice the angle 
 through which the mirror moves. 
 
 Real and Virtual Images. —Real images are those formed 
 by the reflected rays themselves. Virtual images are those 
 formed by the prolongation of the reflected rays. 
 
 Refraction—Laws of — 
 
 1. The sine of the angle of incidence bears to the sine of 
 the angle of refraction a constant ratio for the same two 
 media. 
 
 2. The incident and refracted rays are in the same plane, 
 and that plane is perpendicular to the common surface of 
 the two media. 
 
 N.B .—Refraction is always accompanied by Reflection— 
 the higher the refractive power the greater the amount of 
 Reflection. 
 
 The critical angle of Refraction for water and air is 48V. 
 
 Formula for Spherical mirrors— 
 
 Let /= focal distance ; p = distance of object from mirror ; 
 
 and p = distance of image from mirror, 
 
 1 1 1 1 
 
 then — + — = —r. 
 
 P P / 
 
 Magnitude of Images in Mirrors— 
 
 Length of image: length of object:: distance of image from 
 centre: distance of object from centre. 
 
 Lenses. —Lenses are transparent media, which, from the 
 curvature of their surfaces, have the property of causing the 
 luminous rays which traverse them either to converge or 
 diverge. 
 
42 
 
 c WRINKLES ’ IN ALGEBRA, 
 
 Decomposition of White Light by a Prism .—In the Solar 
 Spectrum there are seven principal colours :—Violet, indigo, 
 blue, green, yellow, orange, and red (v. i. b. g. y. o. r.). The 
 violet is the most, and the red the least refrangible. 
 
 These colours can be recombined to form white light by— 
 
 (a) Taking another prism of the same refracting angle 
 
 as the dispersing one and placing near the other 
 in an inverted position. 
 
 (b) Allowing the decomposed beams to fall upon a 
 
 concave mirror. 
 
 (c) Using Newton’s disc. 
 
 SECTION VII. 
 
 MAGNETISM AND ELECTRICITY. 
 
 A. Magnetism— 
 
 I. Lcavs of Magnetic Force — 
 
 1. Like magnetic poles repel one another; unlike poles 
 attract. 
 
 2. The force exerted between two magnetic poles is pro¬ 
 portional to the product of their strengths, and is inversely 
 proportional to the square of the distance between them. 
 
 II. Differences between Magnets and Magnetic Substances — 
 
 1. Magnets attract only at their poles; they possess 
 opposite properties, and also e magnetic equators'.’ 
 
 2. Magnetic substances are those which attract either 
 pole of the magnet; they have no fixed ‘ poles ’ and no 
 ‘magnetic equators,’ such as iron, steel, nickel, cobalt, and 
 in a much inferior degree chromium and manganese. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 43 
 
 III. Diamagnetic bodies — 
 
 The following are repelled from the poles of a magnet, 
 and are known as diamagnetic bodies:—Bismuth, phos¬ 
 phorus, antimony, and copper. 
 
 IV. Def. of Poles — 
 
 Opposite properties in opposite directions so exactly 
 equal as to be capable of accurately neutralising each 
 other. 
 
 V. Methods of making Magnets — 
 
 1. Single touch. 
 
 2. Divided touch. 
 
 3. Double touch. 
 
 VI. Coercive Force or Retentivity 
 
 Is the force by which a magnetic substance resists the 
 decomposition and the recomposition of the magnetic 
 fluids. 
 
 VII. Magnetic Force may be measured in three ivays — 
 
 1. Balance it against the torsion of an elastic thread. 
 
 2. By observing the time of swing of magnetic needle, 
 oscillating under its influence. 
 
 3. Observe deflection produced in needle, which is 
 already attracted in different direction by a force of known 
 intensity. 
 
 VIII. The earth a Magnet — 
 
 North Magnetic Pole 70° 5' N. Latitude, and 96° 46' 
 W. Longitude. 
 
 South Magnetic Pole has never been reached, but there 
 appear to be two south polar regions. 
 
 IX. Declination — 
 
 The angle between the geographical meridian and the 
 magnetic meridian is called the Declination of that place. 
 
44 ‘ WRINKLES ’ IN ALGEBRA, 
 
 In London now the angle of declination is about 17° 40' 
 West of true N. 
 
 X. Inclination or Dip — 
 
 This is the angle which the magnetic needle makes with 
 the horizon. 
 
 In the north hemisphere the north pole dips; in the 
 south the south pole dips. The inclination at London is 
 about 67° 25'. 
 
 XI. Three things must be Icnoivn to specify the Magnetism of 
 
 any place — 
 
 1. The Declination. 
 
 2. The Inclination. 
 
 3. The Intensity of Magnetic Force. 
 
 XII. Agonic lines are lines of no variation of Declination 
 
 —where the Magnetic and Geographical Meridians 
 coincide. 
 
 XIII. Earth’s Magnetic Equator, or the Aclinic line, is the 
 
 line round the earth upon which the dipping needle 
 remains in a horizontal position. 
 
 XIV. Isogonic lines are lines of equal declination. 
 
 XV. Isoclinic lines are lines of equal dip. 
 
 XVI. Paramagnetic substances are those which are attracted 
 
 by powerful electro-magnets. 
 
 B. Electricity. 
 
 I. Frictional :— 
 
 Laiv /. Like electricities repel, unlike attract. 
 
 Theories of Electricity :— 
 
 A. Symmer’s—Two fluid theory supposes— 
 
 1. There are two imponderable electric fluids of 
 opposite kinds. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 45 
 
 2. They neutralise each other when they combine. 
 
 3. They exist combined in equal quantities in all 
 
 bodies until the condition is disturbed by friction. 
 
 B. Franklin’s Theory supposes— 
 
 1. One fluid only. 
 
 2. When bodies are not neutral, they are charged 
 
 with a positive quantity (+), or a negative 
 quantity ( - ) of the same fluid. 
 
 C. Molecular Theory. — Faraday conceives electrical 
 induction to depend on a physical action between the 
 contiguous molecules, which takes place at a distance by 
 operating through the intervening particles of non-con¬ 
 ducting matter. He also supposes that in this medium 
 successions of layers become alternately, positively and 
 negatively electrified. This condition is called ‘Dielectric 
 Polarisation.’ 
 
 Coulomb's Laivs — 
 
 1. The repulsions and attractions between two 
 
 electrified bodies are in the inverse ratio of the 
 squares of their distances. 
 
 2. Distance being the same, the attractive or re¬ 
 
 pulsive force is directly as the product of the 
 charge. 
 
 Electrical Induction is the action of an electrified body 
 exerted at a distance upon the electricity of another body. 
 
 Inductive Capacity. —The power of a body to allow the 
 inductive influence of an electrified body to act across it 
 is called its Inductive Capacity. 
 
 The Capacity of a Condenser depends upon— 
 
 1. The size and form of the metal plates or coatings. 
 
 2. The thinness of the stratum of insulator between 
 
 them. 
 
 3. The inductive capacity of insulator. 
 
46 
 
 ‘ WRINKLES * IN ALGEBRA, 
 
 B. Electricity— 
 
 II. Current or Dynamical :— 
 
 (а) Definition — 
 
 1. The strength of a current is the quantity of elec¬ 
 tricity which flows past any point of the circuit in one 
 second. 
 
 2. Electromotive-Force (E.M.F.) is that which moves or 
 tends to move electricity from one place to another. 
 
 (б) Volta's Laws — 
 
 I. The difference of potential between any two metals is 
 equal to the sum of the differences of potential between the 
 intervening metals in the contact series. 
 
 II. The total electromotive-force of a series of cells, the 
 zincs and coppers of which are all arranged in one order, 
 will be equal to the electromotive-force of one cell multiplied 
 by the number of cells. 
 
 (c ) Laws of Chemical Action in the Cell — 
 
 I. The amount of chemical action in the cell is propor¬ 
 tional to the quantity of electricity that passes through it— 
 that is, it is proportional to the strength of the current. 
 
 II. The amount of chemical action (in a battery consisting 
 of cells joined in a series) is equal in each cell. 
 
 Joule’s Law. 
 
 (d) Heating Effects of Current — 
 
 The number of units of heat developed in a conductor is 
 proportional 
 
 1. To its resistance. 
 
 2. To the square of the strength of the current. 
 
 3. To the time that the current lasts. 
 
 (e) Ijaws of Electrolysis — 
 
 1. Electrolysis cannot take place unless the electrolyte is 
 a conductor. 
 
ARITHMETIC, EUCLID, MECHANICS, ETC. 
 
 47 
 
 2. The energy of the electrolytic action of the current is 
 the same in all its parts. 
 
 3. The same electric current decomposes chemically 
 equivalent quantities of all the bodies which it traverses. 
 
 4. The quantity of a body decomposed in a given time is 
 proportional to the strength of the current. 
 
 (/) Magnetic Effects of Currents — Ampere's Law — 
 
 To keep movements of a magnet acted on by a current in 
 memory—Suppose a man swimming in the wire with the 
 current, and that he turns so as to face the needle, then 
 the N-seeking pole of the needle will be deflected towards 
 the Left- hand. 
 
 (g) Ohin's Laic — 
 
 The strength of the current varies directly as the electro¬ 
 motive-force, and inversely as the resistance of the circuit. 
 
 This is expressed by formula 
 
 L C = | (1) 
 
 where C = current, E = electromotive-force, R = resistance. 
 
 2. The resistance of a conductor depends on three 
 elements—its conductivity, section, and length. 
 
 cs 
 
 Where l — length, c = conductivity, and 
 
 conductor, so formula (1) now becomes 
 
 E 
 
 C = ~e = C = — 
 
 s = section of 
 
 3. There are two resistances, that between the two plates 
 immersed in liquid and that offered by the conductor which 
 connects the places outside the liquid. Calling the former 
 R and the latter r, the formula becomes 
 
 R + r 
 
 4. Formula for any number (n) of similar cells joined 
 together where the external resistance is very small. 
 
48 
 
 ‘ WRINKLES ’ IN ALGEBRA, ETC. 
 
 C = -?- E 
 
 n\{ ~ E 
 
 5. Formula, when external resistance is great, as in the 
 production of electric light, becomes 
 
 ?zR + r 
 
 if r is very great as compared with tzR, the latter may be 
 neglected and formula becomes 
 
 £ _ wE 
 r 
 
 Laivs of Electro-magnets — 
 
 1. The strength of an electro-magnet is proportional to 
 the strength of the magnetising current. 
 
 2. The strength of an electro-magnet is proportional to 
 the number of turns of wire in the coils. 
 
 3. The strength of an electro-magnet is independent of 
 the thickness and material of the conducting wire. 
 
 4. The strength of an electro-magnet is independent of 
 the diameter of the coils. 
 
 5. A current of electricity requires time to magnetise an 
 iron core to the full extent of its power. 
 
 Laivs of Magneto-electric Induction — 
 
 1. A decrease in the number of lines-of-force which pass 
 through a circuit produces a current round the circuit in 
 the positive direction (i.e., produces a direct current), while 
 an increase in the number of lines-of-force which pass 
 through the circuit produces a current in the negative 
 direction round the circuit. 
 
 2. The total induced electro-motive force acting round a 
 closed circuit is equal to the rate of decrease in the number 
 of lines-of-force which pass through the circuit. 
 
 Len?Js Law .—In all cases of electro-magnetic induction 
 the induced currents have such a direction that their 
 reaction tends to stop the motion which produces them. 
 
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