Such, indeed, is the case, and it is often possible to infer theorems concerning spheres fromtheorems concerning lines, and vice versa.
The point-row and pencil of the second order may be real or imaginary, but his theorems still apply.
This is not to say that many of the theorems and principles involved were not discovered much earlier, but isolated theorems do not make a theory, any more than a pile of bricks makes a building.
All thesetheorems are easy exercises for the student.
The corresponding theoremsin space may easily be obtained by joining the points and lines considered in the plane theorems to a point S in space.
Although their results were obtained by analysis for the most part, nevertheless they have given us theorems which fall naturally into the domain of synthetic projective geometry.
The theorems of this chapter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph.
Taken by themselves, they are of small consequence; it is their relation to other theorems and sets of theorems that gives them their importance.
This theorem is the connecting link between the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry.
The observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding.
In his cranium were financial theorems and syllogisms of the second, third, and fourth power only.
His business as he saw it was with the material facts of life, or, rather, with those third and fourth degree theorems and syllogisms which control material things and so represent wealth.
Mathematicians, though they deduce their theorems from a great height of evidence, yet their first principles are limited by the consideration of Quantity.
Do you mean the principles and theorems of sciences?
It has been observed in another place that the theorems and demonstrations in Geometry are conversant about universal ideas (sect.
From whence, we see that allTheorems too abstruse for Vulgar Apprehensions, which Christianity is believ'd to Teach, however Divine Truths, are yet no part of the Doctrine of Salvation.
And how should this be well brought to pass, but by certain theorems and doctrines; some Concerning the nature of the universe, and some Concerning the proper and particular constitution of man?
Really, all geometrical theoremsare laws of external nature.
Secondly, when the laws of the causes have been found, we calculate the effect of any given combination of them by ratiocination, which may have (though not necessarily) among its premisses the theorems of the sciences of number and geometry.
There are a few theorems connected with this latter subject in Cardan's extraordinary book "On Proportions," but for the most part false and contradictory.
I had indeed capacity enough for it, and learned without difficulty the geometrical and astronomical theorems occurring in the usual course of the school, being well grounded in figures, numbers, and proportions.
It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively.
The preceding theorems are easily adapted to the case of cyclic systems.
We now come to a series of theoremsrelating to the whole motion of the system between any two configurations through which it passes, viz.
Simple illustrations of these theorems are afforded by the chain of straight links already employed.
The theorems of motion just cited are expressed by seven integrals, or equations expressing a law that certain functions of the variables and of the time remain constant.
The general equations expressing the motion of a planet considered as a material particle round a centre of attraction lead to theorems the more interesting of which will now be enunciated.
Pascal relates that de Beaugrand sent all of Roberval's theorems on the cycloid and Fermat's on maxima and minima to Galileo in 1638, pretending that they were his own.
His work is of the extreme of absurdity: he makes a distinction between geometrical and arithmetical fractions, and evolves theorems from it.
We may if we like, for convenience, throw our theorems into the future, and say, e.
The system is evolved in a series of theorems in severely demonstrative order out of the definitions and axioms which we have translated.
The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship.
This extensive theorem of algebraic reciprocity includes many known theorems of symmetry in the theory of Symmetric Functions.
Some of the elementary theorems and various particular problems appear in the works of the earliest algebraists, but the true pioneer of modern researches seems to have been Abraham Demoivre, who first published in Phil.
Beginning, as before, with existence theorems applicable for ordinary values of the variables, we are to consider the cases of failure of such theorems.
Of the first sort are all the theoremsin geometry, asserting an equality of magnitude or ratios, in which the subject and predicate may always change places.
The means which serve to this end, the general good, may be treated astheorems in a geometrical method.
Campanella borrowed his primary theoremsfrom Telesio, but enlarged that Parmenidean philosophy by the invention of his own fertile and imaginative genius.
Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria.
So other theorems of algebra, which are stated as theorems on inequalities, may be regarded as algebraic solutions of problems on maxima and minima.
Extension of the preceding theoremsto the case of relative motions.
We retain several theorems rather because they contain the germs of useful ideas than because of their practical utility, and therefore wish the examiners to restrict themselves scrupulously to the programme.
All the equations which can be obtained by the application of the two theorems relative to quantities of motion and impulsions, reduce themselves to six distinct equations.
Descartes’ theorems upon the intersection of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes.
Analogy of these two theorems with the equations of the equilibrium of a solid, in which the forces are replaced by impulsions and quantities of motion.
General theorems of moments of quantities of motion and impulsions of exterior forces, projected on any axis whatever.
They may be deduced from the preceding rule, but the three last are obtained more simply by extending to a system of material points analogous theorems established for isolated material points.
General theorems of the moments of quantities of motion and impulsions of exterior forces about any axis.
A geometrician, who no longer saw in the world anything but theorems and demonstrations, asked, after the representation of a dramatic masterpiece, "And what does that prove?
He adds, that the existence of God is more certain than the most certain of all the theorems of geometry.
For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity.
The theorems of differential calculus which relate to such functions are in general the same whether the number of variables is two or any greater number, and it will generally be convenient to state the theorems for two variables.
There is no doubt that he arrived at most of his theorems in the first instance by using the method of fluxions.
We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals.
The drawings and machines described in both [Pg022] are exactly the same; but the definitions and theorems quoted above on raising water by fire are not given in the work of Isaac.
Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see T.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "theorems" in a variety of sentences. We hope that you will now be able to make sentences using this word.
