Easy Mathematics, Chiefly Arithmetic: Being a Collection of Hints to Teachers, Parents, Self-taught Students, and Adults, and Containing a Summary Or Indication of Most Things in Elementary Mathematics Useful to be Known

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Macmillan, 1906 - Arithmetic - 436 pages
 

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Page viii - The mathematical ignorance of the average educated person has always been complete and shameless, and recently I have become so impressed with the unedifying character of much of the arithmetical teaching to which ordinary children are liable to be exposed that I have ceased to wonder at the widespread ignorance, and have felt impelled to try and take some step towards supplying a remedy.
Page 293 - If a straight line is divided into any two parts, the sum of the squares on the whole line and on one of the parts is equal to twice the rectangle contained by the whole and that part, together with the square on the other part.
Page 165 - The logarithm of any number to a given base is the index of the power to -which the base must be raised in order to equal the given number. Thus if a' — N, x is called the logarithm of N to the base a.
Page 192 - ... discontinuous. The interspaces are infinitely more extensive than the barriers which partition them off from one another; they are like a row of compartments with infinitely thin walls. All the incommensurables lie in the interspaces; the compartments are full of them, and they are thus infinitely more numerous than the numerically expressible magnitudes. Take any point of the scale at random, that point will certainly lie in an interspace: it will not lie on a division, for the chances are infinity...
Page 194 - On the surface of nature at first we see discontinuity ; objects detached and countable. Then we realise the air and other media, and so emphasise continuity and flowing quantities. Then we detect atoms and numerical properties, and discontinuity once more makes its appearance. Then we invent the ether and are impressed with continuity again. But this is not likely to be the end ; and what the ultimate end will be, or whether there is an ultimate end, is a question difficult to answer.
Page 194 - ... units instead of being dependent on artificial ones, but it throws light also on the nature of phenomena themselves. For instance: The ratio between the velocity of light and the inverted square root of the product of the electric and magnetic constants was discovered by Clerk Maxwell to be i ; and a new volume of physics was by that discovery opened. Dalton found that chemical combination occurred between quantities of different substances specified by certain whole or fractional numbers; and...
Page 191 - ... with incommensurable fractions. Thus only is it that you can deal numerically with such continuous phenomena as the warmth of a room, the speed of a bird, the pull of a rope, or the strength of a current. But how, it may be asked, does discontinuity apply to number? The natural numbers, i, 2, 3, etc., are discontinuous enough, but there are fractions to fill up the interstices; how do we know that they are not really connected by these fractions, and so made continuous again? (By number I always...
Page 194 - ... incommensurable quantities are the rule in physics. Decimals do not in practice terminate or circulate, in other words vulgar fractions do not accidentally occur in any measurements, for this would mean infinite accuracy. We proceed to as many places of decimals as correspond to the order of accuracy aimed at. Whenever, then, a commensurable number is really associated with any natural phenomenon, there is necessarily a noteworthy circumstance involved in the fact, and it means something quite...
Page 194 - I feel inclined to urge that it largely turns on the question as to which way ultimate victory lies in the fight between continuity and discontinuity. On the surface of nature at first we see discontinuity; objects detached and countable. Then we realise the air and other media, and so emphasise conNO.
Page 194 - ... thereto, is a theme not indeed without difficulty but full of importance. It is responsible for the suggestion that energy too may be atomic ! Mendelejeff's series again, or the detection of a natural grouping of atomic weights in families of seven, is another example of the significance of number. Electricity was found by Faraday to be numerically connected with quantity of matter; and the atom of electricity began its hesitating but now brilliant career. Electricity itself — ie electric charge...

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