Theorem

“ A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another. ”

“ Some few writers have sought to distinguish between axiom and postulate by saying that the former was an assumed theorem and the latter an assumed problem, but there is no standard authority for such a distinction, and indeed the difference between a theorem and a problem is very slight. If we say, "A circle may be passed through three points not in the same straight line," we state a theorem; but if we say, "Required to pass a circle through three points," we state a problem. ”

“ Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparison and measurement. But it was found practicable, and, being practicable, was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which are obvious to the most careless observer, and which compose the first principles and ultimate premises of the science. ”

“ The answer is found in some theorem already proved. Turn back to it; perhaps the proof involves some still more fundamental property, and again you ask, why? Again you must recur to some earlier theorem; and so on, until all your "whys" are answered with all possible clearness in irreducible axioms or postulates. He who has the mental ability will find this method of learning a theorem entirely practicable, and it may sometimes be found highly instructive. But it excludes all question of gradual growth of mental power through the successive "stages of culture" itself. ”

“ They are useful as indicating, by one word, the whole enunciation of a theorem.
Every Proposition in the Fifth Book is a Theorem.
PROP. I.—Theorem.
If any number of magnitudes of the same kind (a, b, c, &c.) , be equimultiples of as many others (a′, b′, c′, &c.) , then the sum of the first magnitudes (a + b + c, &c.) shall be the same multiple of the sum of the second which any magnitude of the first system is of the corresponding magnitude of the second system. ”

“ Mathematical necessity, according to the principle of sufficient reason of being, in virtue of which, every relation which is stated in a true geometrical theorem, is as that theorem affirms it to be, and every correct calculation remains irrefutable. ”

“ But, that unpracticed learners, even in making use of one theorem to demonstrate another, reason rather from particular to particular than from the general proposition, is manifest from the difficulty they find in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental power, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem. ”

“ That a Theorem should be 'the formal statement' of itself, has a comfortable domestic sound, something like 'every man his own washerwoman,' but at the same time it involves a fearful metaphysical subtlety. ”

“ He wondered!
"There is no such thing as luck," say the scientifically and mathematically inclined. And, scientifically and mathematically, they demonstrate their theorem utterly to their own satisfaction and that of their ilk. ”

“ Explanation, then, is a kind of classification; it is the finding of resemblance between the phenomenon in question and other phenomena. In Mathematics, the explanation of a theorem is the same as its proof, and consists in showing that it repeats, under different conditions, the definitions and axioms already assumed and the theorems already demonstrated. In Logic, the major [Pg 301] premise of every syllogism is an explanation of the conclusion; for the minor premise asserts that the conclusion is an example of the major premise. ”

“ Thus we already saw that in the mathematician's realm numerous abstract possibilities are excluded whenever a specific theorem is demonstrated. Our rejection, however, of the Third Conception of Being as inadequate was due in the end to a recognition of the fact that, so long as you define mere universals, mere general natures of things, you define neither the Being of objects nor the truth of ideas. ”

“ Our Modulus is the Perfect Man. Our Theorem is the method of use that, by experience and observation everywhere, has been demonstrated as Constructive, enabling the Individual to build toward, and to realize the Modulus. ”

“ It is convenient to mention here the theorem that, x being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n − 1, the coefficients being rational functions of the coefficients of the equation. ”

“ By dropping the use of diagrams, and substituting, in the demonstrations, general phrases for the letters of the alphabet, we might prove the general theorem directly, that is, we might demonstrate all the cases at once; and to do this we must, of course, employ as our premises, the axioms and definitions in their general form. But this only means, that if we can prove an individual conclusion by assuming an individual fact, then in whatever case we are warranted in making an exactly similar assumption, we may draw an exactly similar conclusion. ”

“ It deserves particular notice, that the theorem or speculative truth is not ripe for being turned into a precept, until the whole, and not a part merely, of the operation which belongs to science, has been performed. ”

“ Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. ”

“ By propounding the binomial theorem, the differential calculus, and the integral calculus, he began in 1665 the wonderful series of discoveries in pure mathematics, optics, and physics, which place him in the first rank of the philosophers of all time. ”

“ It does not change one principle involved, or weaken either Modulus or Theorem when we reflect that most equations are ended by death, long before being brought to successful solution. For the time they are certainly interrupted.
Neither do the babel of tongues, the theories, theologies, or philosophies change either Modulus or Theorem, because they are grounded in demonstrated facts, recognized, either vaguely or clearly, in the conscious experience of every intelligent thinking man and woman. ”

“ But the enthymeme is so constantly used in all habitual forms of reasoning, that it does not occur to us as being anything peculiar in mathematical works.
The propositions which are proved to be generally true are termed Theorems: but when anything is required to be done, as to draw a line or a circle under given conditions, this proposition is a Problem. A theorem requires demonstration; a problem, solution. And for both purposes the mathematician usually makes a Construction. ”

“ Such as above described, is the aim and end of the calculus. As for its processes, every one knows that they are simply deductive. In demonstrating an algebraical theorem, or in resolving an equation, we travel from the datum to the quæsitum by pure ratiocination; in which the only premises introduced, besides the original hypotheses, are the fundamental axioms already mentioned—that things equal to the same thing are equal to one another, and that the sums of equal things are equal. ”

“ A theorem is a proposition to be proved, not being able to prove it, we must simply change it according as our experience dictates, this is precisely what we have done with the escapement after having followed the deductions of recognized authorities with the result that we can now illustrate an escapement which has been thoroughly subjected to an impartial analysis in every respect, and which is theoretically and practically correct. ”

“ I have taken a very simple illustration, perhaps the very first theorem which meets the student after those immediately connected with the introductory definitions, both because the simplest illustration is really the best, and because I am here most at home. But the principles of multiple algebra seem to me to shed a flood of light into every corner of the subjects usually treated under the title of determinants, the subject gaining as much in breadth from the new notions as in simplicity from the new notations ”

“ This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n − 1, it is true of n, and thence conclude that it is true for all the whole numbers. [Pg 37]
We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. ”

“ The philosopher peacefully constructs his theorem upon given principles, without deigning to turn his attention to the actual state of .things, or needing the recollection of it to enable him to pursue his inquiry ”

“ The fundamental result in this subject is that every rational integral equation has a root, a theorem which it is by no means easy to prove. It follows without difficulty that an equation of degree n has exactly n roots, real or imaginary. ”

“ The propositions, which consist of both theorems and problems, deal with rectilineal figures, principally the triangle and the parallelogram, and the book concludes with the celebrated Pythagorean theorem and its converse. ”

“ By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c. ”

“ The only one of the premises, therefore, which Art supplies, is the original major premise, which asserts that the attainment of the given end is desirable. Science then lends to Art the proposition (obtained by a series of inductions or of deductions) that the performance of certain actions will attain the end. From these premises Art concludes that the performance of these actions is desirable, and finding it also practicable, converts the theorem into a rule or precept. ”

“ We have now arrived at this step,—that we are in possession of a class of symbols , , , &c. satisfying certain laws, and applicable to the rigorous expression of any categorical proposition whatever. It will be our next business to exhibit a few of the general theorems of the calculus which rest upon the basis of those laws, and these theorems we shall afterwards apply to the discussion of particular examples.
Of the general theorems I shall only exhibit two sets: those which relate to the development of functions, and those which relate to the solution of equations. ”

“ Where theorems are difficult, they should be first taught as exercises in geometrical drawing, until the figure has become thoroughly familiar; it will then be an agreeable advance to be taught the logical connections of the various lines or circles that occur. It is desirable also that the figure illustrating a theorem should be drawn in all possible cases and shapes, that so the abstract relations with which geometry is concerned may of themselves [63] emerge as the residue of similarity amid such great apparent diversity. ”

“ Then it ought to follow that 'the converse of every Theorem of the group will necessarily be true.' Take the converse of the second, i.e.
If 5 < 3, then 5 < 2.
Is this 'necessarily true'? Is every thing which is less than 3 necessarily less than 2?
Nos. Certainly not. I think you have misinterpreted the phrase 'it can be said that one must be true,' when used of the hypotheses. ”

“ Theorem.—The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column. ”

“ His first published paper—a thesis for the Doctorate of Philosophy, in 1799—was devoted to the demonstration that every equation has a root; and of this theorem he made two other distinct demonstrations in 1815 and 1816. ”

“ Not only can no hesitation ever arise respecting the applicability of these principles, since they are true of all magnitudes whatever; but every possible application of which they are susceptible, may be reduced to a technical rule; and such, in fact, the rules of the calculus are. But if the symbols represent any other things than mere numbers, let us say even straight or curve lines, we have then to apply theorems of geometry not true of all lines without exception, and to select those which are true of the lines we are reasoning about. ”

“ Avoiding, however, the mathematical form of expression, one may here try to make clear the metaphysically important nature of theorems of this sort very much as follows: In pure mathematics, the student deals with certain objects that, upon their face, are the products of purely arbitrary definitions. ”

“ Comparatively few of the propositions and proofs in the Elements are his own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him. ”