Ratio

“ X. In general it is clear that, if the value of A increases in a duplicate ratio, yet if the value of B increases in a triplicate ratio, so far from commanding a greater quantity of B, A shall command a smaller quantity; and if A continually goes on squaring its former value, yet if B continually goes on cubing its former value, then, though A will continually augment in value, yet the quantity which it will command of B shall be continually less, until at length it shall become practically equal to nothing. ”

“ Quantities, and the ratio's of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. ”

“ Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another.
First, let A and B have the same ratio to C: A shall be equal to B.
For, if A is not equal to B, one of them must be greater than the other; let A be the greater. ”

“ The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance, is the class of proper fractions. [Pg 72]
In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, the truth is that it is the limit of the corresponding set of rational real numbers in the series of segments ordered by whole and part. For example, is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. ”

“ Socrates: Yes, that is usually what people mean by ratios. The learned people call numbers made from the ratios, rational. Does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers, whether they be odd or even whole numbers?
Boy: You want me to call the numbers made from ratios of whole numbers something called rational? A ratio makes a rational number?
Socrates: Yes boy, can you do that?
Boy: Certainly, Socrates.
Socrates: Do you agree with the way I told him this, Meno? ”

“ Direct and inverse ratios have been applied by an ingenious author to measure human affections, and the moral worth of actions. An eminent Mathematician attempted to ascertain by calculation, the ratio in which the evidence of facts must decrease in the course of time, and fixed the period when the evidence of the facts on which Christianity is founded shall become evanescent, and when in consequence no faith shall be found on the earth.—Reid, Thomas. ”

“ To sum up, there are only relative values; to seek for others is to fall into a contradiction with the very idea of value in exchange which necessarily implies the idea of a ratio between two terms. But also an accomplished change in this ratio is a relative effect, which can and should be explained by absolute changes in the terms of the ratio. There are no absolute values, but there are movements of absolute rise and fall in values. ”

“ Value being a ratio, it is impossible for all values to rise or fall simultaneously. The sum of subjective values may increase or decrease,—indeed it is one of the great objects of human endeavour to increase the sum of want-satisfying power,—but the sum of the ratios between these subjective values is constant. ”

“ If the quantity of purchasable articles increases while the quantity of money remains the same, the value of the money increases in the same ratio. If the quantity of money increases while the quantity of purchasable articles remains the same, the value of money decreases in the same ratio. ”

“ X.12 We choose, then, as one of the best index numbers of prices, the average price of the goods sold, those goods being measured in units worth a dollar in the base year; in other words, the ratio of the value of sales at actual prices to the value of the same sales if made at base prices; in still other words, the weighted arithmetical average of all price ratios, each ratio being weighted according to the values sold, reckoned at base prices. ”

“ If in comparing indetermined quantities of different sorts with another, any one is said to be as any other directly or inversly: the meaning is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. and if any one is said to be as any other two or more directly or inversly: the meaning is, that the first is augmented or diminished in the ratio's in which the others, or the reciprocals of the others, are augmented or diminshed. ”

“ The ratio which a token of value, whether made of paper or of debased gold or silver, bears to certain weights of gold or silver estimated according to the mint price, depends not on its own composition but on the quantity in which it is found in circulation. ”

“ When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150 : 200 is not the same as 100 : 150. The mean ratio is as unity to √1·00 × 2·00 or 1 to 1·41. ”

“ 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. ”

“ No! We are mere prosaic mortals: it is beyond us!
In p. 49 I see some strange remarks about ratios. First look at Def. 44. 'When a magnitude is contained an exact number of times in two magnitudes of its kind, it is said to be their common measure.' (The wording is awkward, and suggests the idea of their having only one 'common measure'; but let that pass.) 'The ratio of two magnitudes of the same kind is the number which would express the measure of the first, if the second were taken as unity.'
'The measure of the first'! Do you understand that? ”

“ Section 39 If the first two terms are added together and the last two terms are added together, the ratio of the sum of the first addition to the first term is equivalent to the ratio of the sum of the second addition to the third term. ”

“ Abstractly, it is the ratio at which commodities may be exchanged for each other, or, since such ratio for a unit of one4 commodity is expressed by the amount of another given for it, the exchange value of a thing is the quantity of some other thing that will be evenly exchanged for it, or, considered in a general sense, the amount of commodities in general it will exchange for,—its general purchasing power, in short. ”

“ To all cases of geometrical progression the application is obvious for finding the common ratio, the number of terms, or any particular term in the series. An approximate solution is thus given to all questions of compound interest, to regularly progressive increase of population, and to many calculations of chances. ”

“ Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as for example 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculæ, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a301,030 to 1 and so on ”

“ If one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable, in a series prescribed by the ratio to the powers of this variable, it is easy to see that in supposing the variable equal to unity the fraction becomes 1⁄2, and the series becomes plus one, minus one, plus one, minus one, etc. ”

“ The proportion consecrated may be a certain ratio of income fixed on a sliding scale, on the principle that the greater the profits, the greater the proportion which me be spared. For instance, on the first day of each week, or month, or quarter, or year, one may consecrate a certain proportion of his profits of that week, month, quarter, or year to the Lord, say five, eight, or ten per cent., in case they rise to a specified amount ”

“ In cases where prices, or exchange relations, are not in accord with values, the term "ratio of exchange" is inapplicable, since there are no quantities to be terms of the ratio—except the pure abstract numbers of the commodities, each measured in its own unit, exchanged. ”

“ The ratio between the price which he has to pay for these means of production, and the produce which they enable him to raise, is the rate of his profit. If he must give for labour and tools four-fifths of what they will produce, the remaining fifth will constitute his profit, and will give him a rate of one in four, or twenty-five per cent, on his outlay. ”

“ If the income distribution remains the same (all incomes grow with the same rate) then also the average income, y = W grows at the same rate, and then also the net income distribution remains the same, and then the ratio of net average to subsistence remains the same too. ”

“ The unchecked increase of population is in a geometrical ratio, the increase of food for their subsistence is in an arithmetical ratio. That is, while humankind would increase in proportion as 1, 2, 4, 8,16, 32, 64, 128, 256, food would only increase as 1, 2, 3,4, 5, 6, 7, 8, 9. The more the mouths the less the proportion of food. ”

“ Give me the statistics of pyrotechnic powder burnt by a people, and I shall tell you the standard measure of their souls and bodies. If the figure be a maximum, then the physical and moral measure will be the minimum, for the ratio is inverse. ”

“ Also, we see that is the numerical value of the ratio between the height of any ordinate and that of the next preceding it. In Fig. 40, we have taken as ; each ordinate being as high as the preceding one.
If two successive ordinates are related together thus in a constant ratio, their logarithms will have a constant difference; so that, if we should plot out a new curve, Fig. ”

“ Perhaps it is within the ‘Senior Wrangler’s’ knowledge of physical exploration, that when the physicist finds a result not bearing that ratio to its assigned cause which the two were ascertained in other cases to have, he immediately assumes the presence of some perturbing cause or causes, which modify the ratio. ”

“ When gold replaced silver as a measure of value, the same name was applied according to the ratio between the values of silver and gold, to perhaps 1-15th of a pound of gold. The word pound, as a money-name, thus becomes differentiated from the same word as a weight-name. [8] (3) The debasing of money carried on for centuries by kings and princes to such an extent that, of the original weights of the coins, nothing in fact remained but the names. ”

“ When is a line divided harmonically?
21. When a line is divided harmonically, what are corresponding pairs of points called? Ans. Harmonic conjugates.
22. What are reciprocal ratios?
23. Give one enunciation that will include Propositions xxii., xxiii. of Book V.
Exercises on Book V.
Def. I.—A ratio whose antecedent is greater than its consequent is called a ratio of greater inequality; and a ratio whose antecedent is less than its consequent, a ratio of lesser inequality. ”

“ Is it in contemplation with the House of Representatives to extend a like improvement to our measures and weights, and to arrange them also in a decimal ratio? The facility which this would introduce into the vulgar arithmetic would, unquestionably, be soon and sensibly felt by the whole mass of the people, who would thereby be enabled to compute for themselves whatever they should have occasion to buy, to sell, or to measure, which the present complicated and difficult ratios place beyond their computation for the most part. ”

“ 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. ”