190 THIRD REPORT — 1833. 



to the exclusion of symbolical algebra, as the only form of it 

 which was capable of strict demonstration, and which alone, 

 therefore, was entitled to be considered as a science of strict and 

 logical reasoning. The arguments which they made use of 

 were unanswerable, when advanced against the form under 

 which the principles of algebra were exhibited in the elemen- 

 tary and all other works of that period, and which they have 

 continued to retain ever since, with very trifling and unimpor- 

 tant alterations ; and the system of algebra which was formed 

 by the first of these authors was perfectly logical and complete, 

 the connexion of all its parts being capable of strict demon- 

 stration; but there were a great multitude of algebraical re- 

 sults and propositions, of unquestionable value and of unques- 

 tionable consistency with each other, which were irreconcila- 

 ble with such a system, or, at all events, not deducible from it ; 

 and amongst them, the theory of the composition of equations, 

 which Harriot had left in so complete a form, and which made 

 it necessary to consider negative and even impossible quan- 



learner are employed, or some arts are used wliicli are not justifiable. The 

 first error in teaching the first principles of algebra is obvious on perusing a few 

 pages only of the fii-st part of Maclaurin's Algebra. Numbers are there divided 

 into two sorts, positive and negative : and an attempt is made to explain the 

 nature of negative numbers, by allusions to book debts and other arts. Now 

 when a person cannot explain the principles of a science, without reference to 

 a metaphor, the probability is, that he has never thought accurately upon the 

 subject. A number may be greater or less than another number : it may be 

 added to, taken from, multiplied into, or divided by, another number ; but in 

 other respects it is very intractable; though the whole world should be destroyed, 

 one will be one, and three will be three, and no art whatever can change their 

 nature. You may put a mark before one, wjiich it will obey ; it submits to be 

 taken away from another number greater than itself, but to attempt to take it 

 away from a number less than itself is ridiculous. Yet this is attempted by 

 algebraists, who talk of a number less than nothing, of multiplying a negative 

 number into a negative number, and thus producing a positive number, of a 

 number being imaginary. Hence they talk of two roots to every equation of 

 the second order, and the learner is to try which will succeed in a given equa- 

 tion : they talk of solving an equation which requires two impossible roots to 

 make it soluble : they can find out some impossible numbers, which being 

 multiplied together pi-oduce unity. This is all jargon, at which common sense 

 recoils ; but from its having been once adopted, like many other figments, it 

 finds the most strenuous supporters among those who love to take things upon 

 trust and hate the colour of a serious thought." 



" From the age of Vieta, the father, to this of Maseres, the restorer of alge- 

 bra, many men of the greatest abilities have employed themselves in the pursuit 

 of an idle hypothesis, and have laid down rules not founded in truth, nor of any 

 sort of use in a science admitting in every step of the plainest principles of 

 reasoning. If the name of Sir Isaac Newton appears in this list, the number 

 of advocates for errour must be considerable. It is, however, to be recollected, 

 that for a much longer period, men scarcely inferiour to Newton in genius, and 

 his equals, probably, in industry, maintained a variety of positions in philoso- 



