Library of Useful Knowledge: Mathematics I.Baldwin and Cradock, 1836 |
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Side 16
... substitution will not materially affect the result . We will now proceed to shew how to find a series of decimal fractions , which approach nearer and nearer to a given fraction , and also that , in this approximation , we may approach ...
... substitution will not materially affect the result . We will now proceed to shew how to find a series of decimal fractions , which approach nearer and nearer to a given fraction , and also that , in this approximation , we may approach ...
Side 22
... substituted for a and b . If we place 6 and 8 instead of a and b , we shall find that the two ex- pressions are not equal , and therefore we must conclude that they are not always the same . Thus in the expres- sions 3 x 4 and 2 x + 8 ...
... substituted for a and b . If we place 6 and 8 instead of a and b , we shall find that the two ex- pressions are not equal , and therefore we must conclude that they are not always the same . Thus in the expres- sions 3 x 4 and 2 x + 8 ...
Side 23
Mathematics I. may be substituted for one of the letters , provided care is taken to make the sub- stitution wherever that letter occurs thus from the formula : a2 — b2 = ( a + b ) ( a - b ) we may deduce the following by making ...
Mathematics I. may be substituted for one of the letters , provided care is taken to make the sub- stitution wherever that letter occurs thus from the formula : a2 — b2 = ( a + b ) ( a - b ) we may deduce the following by making ...
Side 44
... substituted in- stead of x , the result is 4a 2a3 = 2a2 × 4a2 ; if a be substituted instead of x , the result is 0 = 0 , which show that + a and a are both correct values of x . We have here noticed , for the first time , an equation of ...
... substituted in- stead of x , the result is 4a 2a3 = 2a2 × 4a2 ; if a be substituted instead of x , the result is 0 = 0 , which show that + a and a are both correct values of x . We have here noticed , for the first time , an equation of ...
Side 45
... substituted for x , makes y m and n are the same as in the first and third . 1 In the first and third , m is a 0 , and in the second and fourth ist . y = ax2 bx + c = a ( x — m ) ( x − n ) m + n = -- y = ax2 + bx + c = a ( x + m ) ( x ...
... substituted for x , makes y m and n are the same as in the first and third . 1 In the first and third , m is a 0 , and in the second and fourth ist . y = ax2 bx + c = a ( x — m ) ( x − n ) m + n = -- y = ax2 + bx + c = a ( x + m ) ( x ...
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algebra algebraical quantities answer arithmetic ax² becomes binomial theorem called ciphers coefficient common logarithms contained cube root decimal fraction decimal places decimal point digits divided dividend division divisor equa equal equation example exponent expression factors figure frac geometry given gives greater inches instance least common multiple less letters loga logarithm magnitude means metical multiply negative nth root observe operations permutations positive preceding proceed proportion proposition proved quotient reasoning reduced remainder result right angles rithm rule shillings solution square root student subtract suppose theorem things taken third tion triangle true unit unity unknown quantities vulgar fraction whole number write yards
Populære avsnitt
Side 69 - Parallelograms on the same or equal bases, and between the same parallels, are equal. The explanation of this is as follows : the whole proposition is divided into distinct assertions, which are placed in separate consecutive paragraphs, which paragraphs are numbered in the first column on the left ; in the second column on the left we state the reasons for each paragraph, either by referring to the preceding paragraphs from which they follow, or the preceding propositions in which they have been...
Side 12 - D' contains £T of G. Here then appears a connexion between the multiplication of whole numbers, and the formation of a fraction whose numerator is the product of two numerators, and its denominator the product of the corresponding denominators. These operations will always come together, that is whenever a question occurs in which, when whole numbers are given, those numbers are to be multiplied together ; when fractional numbers are given, it will be necessary, in the same case, to multiply the...
Side 53 - ... with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given, of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound.
Side 70 - Thus, that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, was an experimental discovery, or why did the discoverer sacrifice a hecatomb when he made out its proof ?
Side 3 - ... faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind...
Side 84 - When it is said that the angle = arc , it is only meant that, is equal to the radius, the number of these units in any other angle is found by dividing the number of linear units in its arc by the number of linear units in the radius. It only remains to give a formula for finding the number of degrees, minutes, and seconds in an angle, whose theoretical measure is given. It is proved in geometry that the ratio of the circumference of a circle to its diameter, or that of half the circumference to...
Side 13 - This process is then, by extension, called division : y is called the quotient of £ divided by J., and is found by multiplying the numerator of the first by the denominator of the second for the numerator of the result, and the denominator of the first by the numerator of the second for the denominator of the result. That this process does give the same result as ordinary division in all cases where ordinary division is applicable, we can easily shew from any two whole numbers, for example, 12 and...
Side 80 - In each succeeding term the coefficient is found by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing by the number of the preceding term.
Side 3 - ... thought advisable to make many passages between ports that were well known before attempting a voyage of discovery. So it is with our reasoning faculties : it is desirable that their powers should be exerted upon objects of such a nature, that we can tell by other means whether the results which we obtain are true or false, and this before it is safe to trust entirely to reason. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds : 1. Every term is distinctly...
Side 74 - ... number of combinations of n things r at a time is the same as the number of combinations of n things n — r at a time ; This result is frequently useful in enabling us to abridge arithmetical work.