It is seen at once that this result is a generalization of the theorem for r = 1, and its proof is conveniently given by induction from that case.
The fundamental theorem in regard to the rationality group.
Then the theorem is that there exists one, and only one, set of functions z1, .
For the case when the number of equations is the same as that of dependent variables, the following is a general theorem which should be referred to: Let there be r equations in r dependent variables z1, .
The proof of this theorem may be given by showing that if ordinary power series in x1 - -a1, .
In the classic dynamics we use, to find these elementary domains, the theorem that two physical states of which one is the necessary effect of the other are equally probable.
The theorem that the square of the diagonal is double the square of the side--that famous discovery of primitive mathematics, in honour of which the legendary Pythagoras is said to have sacrificed a hecatomb--is elicited from him.
Wherfore saith the Theorem that the long square made of D.
But as they be irregulare formes, that is to saye, suche formes as haue no precise measure nother proportion in their draughte, so can there scarcely be made any certaine theorem of them.
Theorem, so in the same Theorem it is declared, y^t their ground lines are equall togither, that is C.
Weights imply scales, of which we have also mention; and scales involve the primary theorem of mechanics in its least complicated form--involve not a qualitative but a quantitative prevision of mechanical effects.
As already shown, the first theorem established in mechanics was, that equal weights suspended from a lever with equal arms would remain in equilibrium.
The circumference and area of a circle could be computed from its radius, and the Pythagorean theorem related the three sides of a right triangle.
The theorem of "national genius" will suffice to impair any exposition, however judicious otherwise.
This the mathematician instantly sees is equivalent to the theorem that the circumference of any circle is double of the diagonal of the cube on its diameter.
Wilson did not publish anything: the theorem by which he has cut his name in the theory of numbers was communicated to Waring, by whom it was published.
Iamblichus, the disciple of Porphyry (died 330), transformed Neoplatonism "from a philosophic theorem into a theological doctrine.
Gnosticism here as frequently elsewhere is related to common Christianity as excess progressing to the invention of a myth with a tendency to a historical theoremdetermined by the effort to maintain one's own position; cf.
The most important theorem in regard to partial differential coefficients is the theorem of the total differential.
An important theorem, first proved by Euler, is immediately deducible from the theorem of the total differential.
This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known.
In the Quadratura (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically.
There is a corresponding theorem for a closed curve in two dimensions, viz.
An important application of the theorem of intermediate value and its generalization can be made to the problem of evaluating certain limits.
The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis.
He had stretched Wysocki's theorem too far and it had failed.
He who has the mental ability will find this method of learning a theorem entirely practicable, and it may sometimes be found highly instructive.
Open the book at the middle; there you may find a theorem whose proof you readily understand, because it implies very little previous knowledge.
The answer is found in sometheorem already proved.
He did the world great service in the discovery of the so-called Pythagorean theorem in geometry, that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.
An important theoremcontained in it is known as Green's theorem, and is of great value.
As a consequence of Fourier's theorem it follows that any periodic curve having any wave form can be imitated by the superposition of simple sine currents differing in maximum value and in phase.
This theorem enables us, when one funicular has been drawn, to construct any other without further reference to the force-diagram.
The above construction for Z is a corollary of the general theorem given in S 127.
By means of this theorem we can show that the previous reduction of any system to a wrench is unique.
If we put x, y, z = 0, the theoremis proved as regards axes parallel to Ox.
The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other.
It follows from Euler's theorem that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position O, followed by a pure rotation about some axis through O.
By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated.
The proof is similar to that of the correspondingtheorem of plane kinematics (S 3).
It follows from Clairaut's theorem that if the earth is an oblate spheroid, its ellipticity can be determined from relative values of gravity and the absolute value at the equator involved in c.
In 1849, Stokes derived a theorem whereby the distance N of the geoid from the ellipsoid of reference can be obtained from an integration of gravity anomalies over the surface of the earth.
Observations with nonreversible, invariable compound pendulums have contributed to the application of Clairaut's theorem in its original and contemporary extended form for the determination of the figure and gravity field of the earth.
Huygens derived a theorem which has provided the basis for the employment of the reversible compound pendulum for the absolute determination of the intensity of gravity.
Then Clairaut, from the assumption that the earth is a spheroid of equilibrium, derived a theorem from which the ellipticity of the earth can be derived from values of the intensity of gravity.
When the times were judged to be the same, the distance between the knife edges was inferred to be the length of the equivalent simple pendulum, in accordance with Huygens' theorem on conjugate points of a compound pendulum.
The theorem is that a given compound pendulum possesses conjugate points on opposite sides of the center of gravity; about these points, the periods of oscillation are the same.
It deserves particular notice, that thetheorem 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.
The theorem is not stated by Laplace in the exact terms in which I have stated it; but the identity of import of the two modes of expression is easily demonstrable.
The enunciation, as it is called, that is, the general theorem which stands at the head of the demonstration, is not the proposition actually demonstrated.
The science receives it, considers it as a phenomenon or effect to be studied, and having investigated its causes and conditions, sends it back to art with a theorem of the combination of circumstances by which it could be produced.
Pythagoras possessed immense knowledge; he invented the theorem of the square of the hypothenuse, and he first divided the year into 365 days and 6 hours.
A modern historian aptly remarks that the medicine of the present "embraces nothing but a theorem of investigation by the senses.
Proof of the Fundamental Theorem of Invariants; Philosophic Magazine (1878), p.
I remember, one bright pupil, who, after I had stated and demonstrated this theorem to him, and he had caught the full import of it, was moved as by a miracle.
If an algebraic theorem can be interpreted in arithmetic, the corresponding arithmetical theorem is therefore true.
No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.
Newton was indebted to it for his theorem of the binomial and the principle of universal gravity.
After Pythagoras discovered his fundamental theorem he sacrificed a hecatomb of oxen.
Illustration: A geometrical drawing including square and four triangles to demonstrate a graphical proof of thetheorem of Pythagoras as described in the poem.
Fourier's theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.
Professor Cayley has since informed me that thetheorem about whose origin I was in doubt, will be found in Schlaefli's "De Eliminatione.
For the same reason, Ricardo's theorem would be inadequate 'sociologically,' or as a formula which would enable us to predict the future distribution of wealth.
According to his disciple, M'Culloch, Ricardo's great merit was that he 'laid down the fundamental theorem of the science of value.
What theorem allows you to change any proportion into an equation?
What theorem allows you to change any equation into a proportion?
Reference:~ The chapter on The Binomial Theoremin any algebra.
Write out the proofs for the following, stating the theorem in full in each case: (a) The product of the extremes equals etc.
Our theorem is, then, demonstrated both by facts and by intrinsic reasons.
The theorem is this: All efficient cause is infinitely more perfect, and of an infinitely better nature, than any of its effects.
As many other points as may be desired in the curve NC are found by a Theorem which Mr. Barrow has demonstrated in section 12 of his Lectiones Opticae, though for another purpose.
In Mathematics, the explanation of a theoremis 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 Geometry, the proof of a theorem depends not only upon the definition of the figure or figures directly concerned, but also upon one or more axioms, and upon propria or constructions already established.
If there are no such things as isosceles triangles, Euclid's fifth Proposition is only formally true, like a theoremconcerning the fourth dimension of space: merely consistent with his other assumptions.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "theorem" in a variety of sentences. We hope that you will now be able to make sentences using this word.
