Proportion (mathematics)

“ Though proportion be measurable in its own nature, and therefore hath proper quantity, yet as things cannot have proportion which have not quantity of some other kind, it follows that whatever has quantity must have it in one or other of these three kinds, extension, duration, or number. ”

“ Some proportion is also required in different things intimately connected, as betwixt a dwelling-house, the garden, and the stables. But we require no proportion among things slightly connected, as betwixt the table a man writes on and the dog that follows him.{392} Proportion and uniformity never coincide: things perfectly similar are uniform; but proportion is never applied to them: the four sides and angles of a square are equal and perfectly uniform; but we say not that they are proportional. Thus, proportion always implies inequality or difference ”

“ And the Circle, then the Quadrate. [40]
16. If prime figures be of equall height, they are in reason one unto another, as their bases are: And contrariwise.
The proportion of first figures is here twofold; the first is direct in those which are of equall height. In Arithmeticke we learned; That if one number doe multiply many numbers, the products shall be proportionall unto the numbers, multiplyed. From hence in rationall figures the content of those which are of equall height is to bee expressed by a number. ”

“ The fifth view is nothing more than a change of words: A number reduced to 7/9 of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation 7/9 of a = 7 times a/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion.
When the student comes to algebra, he will find that, in all the applications of that science, fractions such as a/b most frequently require that a and b should be themselves supposed to be fractions. ”

“ One arithmetical proportion is the same with another arithmetical proportion, when one of the antecedents exceeds its consequent, or is exceeded by it, as much as the other antecedent exceeds its consequent, or is exceeded by it. And therefore, in four magnitudes, arithmetically proportional, the sum of the extremes is equal to the sum of the means. For if A. B :: C. D be arithmetically proportional, and the difference on both sides be the same excess, or the same defect, E, then B + C (if A be greater than B) will be equal to A - E + C ”

“ It would indeed be a fatal mistake to suppose that proportion cannot be susceptible of great variety, since the whole meaning of the term has reference to the adjustment and proportional correspondence of variable properties. ”

“ It appears at first paradoxical, but it is well known to all mathematicians, that, through all eternity, fractions may be taken from a weight without the weight ever being annihilated. It is sufficient that each successive fraction be less than the preceding one, in a determined and regular proportion. ”

“ A short rule may be found for adding together any number of terms of a continued proportion. Let it be first required to add together the terms 1, r, rr, &c. where r is greater than unity. It is evident that we do not alter any expression by adding or subtracting any numbers, provided we afterwards subtract or add the same. ”

“ Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to one another and to the whole; so that it may be increased by addition of like parts, and diminished by subtraction, may be multiplied and divided, and, in a word, may bear any proportion to another quantity of the same kind, that one line or number can bear to another. That this is essential to all mathematical quantity, is evident from the first elements of algebra, which treats of quantity in general, or of those relations and properties which are common to all kinds of quantity. ”

“ If any two unequal quantities be made extremes, and there be interposed betwixt them any number of means in geometrical proportion, and the same number of means in arithmetical proportion, the several means in geometrical proportion will be less than the several means in arithmetical proportion. ”

“ Amongst the various difficult questions which are always cropping up in connection with our art is: What is proportion? and on this topic we give the definition of proportion as laid down by Virtrivius. "Proportion is the commensuration of the various constituent parts with the whole, on the existence of which symmetry is found to consist." As, for instance, the human frame is divided into four equal parts by very distinctly marked divisions in its structure and outward form. ”

“ When we first treat numerical results in any novel kind of investigation, our impression will probably be that one quantity varies in simple proportion to another, so as to obey the law y = mx + n. We must learn to distinguish carefully between the cases where this proportionality is really, and where it is only apparently true. In considering the principles of approximation we found that a small portion of any curve will appear to be a straight line. When our modes of measurement are comparatively rude, we must expect to be unable to detect the curvature. ”

“ Magnitudes which have the same ratio are called proportionals. When four magnitudes are proportionals, it is usually expressed by saying, “The first is to the second as the third is to the fourth.”
viii. Analogy or proportion is the similitude of ratios.
We have given the foregoing definitions in the order of Euclid, as given by Simson, Lardner, and others;2 2Except that viii. is put before vii., because it relates, as v. and vi., to the equality of ratios, whereas vii. is a test of their inequality. but it is evidently an inverted order; for vi. viii. are definitions of proportion, and v. ”

“ There is no reason why the mathematical fact should be what it is except the will of the Being who imposed the law. Any other proportion could equally well be expressed mathematically, and its results calculated. ”

“ If one proportion be given, from it an indefinite number of other proportions can be inferred, and a great part of the theory of proportion consists in proving the truth of these derived proportions. Geometers make use of certain technical terms to denote the most important of these processes. ”

“ Division is defined as the operation inverse to multiplication; but begin by an example taken from the familiar notion of partition and show on this example that multiplication reproduces the dividend.
There still remain the operations on fractions. The only difficulty is for multiplication. It is best to expound first the theory of proportion; from it alone can come a logical definition ”

“ If two transmitted lengths have to their times any other proportion explicable by number, and the side A B be so divided in E, that A B be to A E in the same proportion which the lengths transmitted have to the times in which they are transmitted, and between A B and A E there be taken a mean proportional A F ”

“ I would answer, that this is a question to which we can never give any precise answer; nor is it possible to reduce to numbers the proportion, which we ought to fix betwixt them. One may safely affirm, that this proportion finds itself without any art or study of men; as we may observe on many other occasions. ”

“ The power which is given by such an analysis, of expressing with more ease the mathematical laws of phenomena, depends in general on this, that since the calculus applies, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments, we can always substitute for each increment any other magnitude more easy to consider, provided that their final ratio is the ratio of equality, or, in other words, that the limit of their ratio is unity. ”

“ It will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geometric magnitudes, while the ancient treatment is independent of this theory. ”

“ The fifth book stands alone, depending upon none of the preceding books, and contains the theory of proportion, with respect not merely to geometrical magnitudes, such as lines, angles, areas, solids, but to any magnitudes of which multiples can be formed. The diagrams consist of straight lines, probably for convenience of construction, but the enunciations of the propositions and the reasoning are perfectly general. With the exception of his treatment of parallels, Euclid's doctrine of proportion has been the subject of more discussion than any other part of the Elements. ”

“ If it were not for this opinion, it had been impossible for the doctrine of proportion to have held its ground very long; the world would be soon weary of hearing of measures which related to nothing, either of a natural principle, or of a fitness to answer some end; the idea which mankind most commonly conceive of proportion, is the suitableness of means to certain ends, and, where this is not the question, very seldom trouble themselves about the effect of different measures of things. ”

“ Absolute equality is not required, still less absolute similarity.
Symmetry must not be confounded with Proportion. Symmetry is the opposition of equal quantities; proportion is the due connection of unequal quantities with each other. A tree, in sending out equal boughs on opposite sides, is symmetrical; in sending out smaller boughs toward the top, proportional. In the human face its balance of opposite sides is symmetry ”

“ It will be seen that if we should judge by these four balls of the proportion in the urn, 32 times out of 81 we should find it 1⁄4, and 24 times out of 81 we should find it 1⁄2, the truth being 1⁄3. To extend this table to high numbers would be great labor, but the mathematicians have found some ingenious ways of reckoning what the numbers would be. It is found that, if the true proportion of white balls is p, and s balls are drawn, then the error of the proportion obtained by the induction will be ”

“ The celebrated interminable fraction 3.14159..., which the mathematician calls , is the ratio of the circumference to the diameter. But it is thousands of things besides. It is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention. ”

“ That relations of proportion enter into beautiful form is allowed by all. A technically trained eye may recognize, and perhaps enjoy, simple numerical ratios among magnitudes in lines, etc., but this factor does not appear to enter, in a conscious way at least, into ordinary æsthetic appreciation of form. We hardly experience any addition of enjoyment in learning that the ratio of the axes of a pleasing oval is 2 : 1. So far as conscious reflection can tell us, our enjoyment of proportion rests on a vague estimation of one magnitude in relation to another. ”

“ May not a man who is a mathematician as distinctly conceive that in the infinite variety of numbers gradually ascending, there is no ratio whatsoever which is not equal to the ratio of some one whole number to some other whole number? Yet the mathematician can demonstrate that there are innumerable ratios which are not equal to the ratio of some one number to another. Many mathematicians, taking it for granted that it was possible to square the circle, have spent their lives in a fruitless pursuit. ”

“ Propor′tionless, without proportion or symmetry of parts.—Proportional representation, representation in a parliament, &c., according to the number of electors in an electoral district.—Compound proportion, a proportion between two quantities, each of which is the product of two or more, those of the first set being in their order proportional to those of the other; In proportion, in the degree or measure, according; Inverse, or Reciprocal, proportion, an equality of ratio between two quantities and the reciprocals of other two, as 6 : 3 = 1⁄2 : 1⁄4. ”

“ It must be remembered that the number expressing any quantity in terms of a unit is inversely proportional to the magnitude of the unit, i.e. the number of new units is to be found by multiplying the number of old units by the ratio of the old unit to the new unit. ”

“ This similitude consisteth in two things, to witt in the equality of their angles, and proportion of their shankes.
Therefore,
20. Like figures have answerable bounds subtended against their equall angles: and equall if they themselves be equall.
Or thus, They have their termes subtended to the equall angles correspondently proportionall: And equall if the figures themselves be equall; H. This is a consectary out of the former definition.
And
21. ”

“ The great general principle of Proportion needs no special illustration or exercise, because it is so intimate a part of all other principles and exercises. It may be studied in every example of supreme art. It is the foundation of all the finest work in line and mass. The mystery of Spacing will be revealed to the mind that has developed Appreciation.
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SYMMETRY. The most common and obvious way of satisfying the desire for order is to place two equal lines or shapes in exact balance, as in a gable, windows each side of a door, or objects on a shelf. ”

“ But beside the objection of the “double emploi,” it is very certain that the method of reduction to unity presents, in their true light and in a complete and simple manner, all the questions of ratio which are the bases of arithmetical solutions; so that the subsequent introduction of proportions teaches nothing new to the pupils, and only presents the same thing in a more complicated manner. We therefore exclude from our programme of examination the solution of questions of arithmetic, presented under the special form which constitutes the theory of proportions. ”