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Updated: June 22, 2025
The propositions of Arithmetic affirm the modes of formation of given numbers, and are true of all things under the condition of being divided in a particular way. Algebraical propositions, on the other hand, affirm the equivalence of different modes of formation of numbers generally, and are true of all things under condition of being divided in any way.
The first, if they condescend to love you, love you as the biggest doll they have yet dandled, and for a doll's good qualities, your pretty blue eyes and your exquisite millinery. The last, if they prudently accept you, do so on algebraical principles; you are but the X or the Y that represents a certain aggregate of goods matrimonial, pedigree, title, rent-roll, diamonds, pin-money, opera-box.
Since then algebraical truths are true of all things whatever, and not, like those of geometry, true of lines only or angles only, it is no wonder that the symbols should not excite in our minds ideas of any things in particular.
It was quite enough if one of the two knew anything about the matter; besides, he thought that it improved me so much to look over the problems and algebraical calculations of my schoolfellows. With this man I was continually measuring my strength; and as I conceived that I found, myself woefully wanting, he proved an excellent moral sedative to my else too rampant vanity.
In the same general manner, mechanics, astronomy, and in a less degree, every branch of natural philosophy commonly so called, have been made algebraical.
Nevertheless, it will appear on consideration, that this apparently so decisive instance is no instance at all; that there is in every step of an arithmetical or algebraical calculation a real induction, a real inference of facts from facts; and that what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of the language.
Let us assume that the two teachers are on a level in respect of their capacity for influencing and stimulating their pupils, and let us indicate that level by the algebraical symbol x.
"The first friend I ever cared for was a boy marvellously endowed for the study of mathematics. At the time of our first meeting at school, I was unable to solve even the simplest algebraical problem. But we had been together only for half a month, when we exchanged parts.
Since then algebraical truths are true of all things whatever, and not, like those of geometry, true of lines only or of angles only, it is no wonder that the symbols should not excite in our minds ideas of any things in particular.
In the same general manner, mechanics, astronomy, and in a less degree, every branch of natural philosophy commonly so called, have been made algebraical.
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