80J 



PROPORTION 1 . 



PROPYL. 



810 



long and intricate propositions, we should agree with those who woul 

 refuse to place those propositions before beginners, on the ground tha 

 unimpeachable demonstration is already given, namely, an assumption 

 which those who would rather dispense with it do not deny to be botl 

 true and easy (or capable of being made easy), and logic which is un 

 contested and incontestable. But the vice of the system which is sub 

 stituted for that of Euclid, consists in the entrance of an assumptior 

 which is not true ; for to reason upon all magnitudes as if they were 

 commensurable, and to assert conclusions derived from such reasonings 

 is to assume that all magnitudes are commensurable, which is not true 

 The method of Lacroix, as above explained, is sound as far as it goes 

 he asserts that the propositions of geometry are sensibly true, or i 

 false, imperceptibly false, and this he proves. But he makes geometr^ 

 cease to be an exact science. That of Legendre, on the other hand 

 though it proves no more than that of Lacroix, professes to prove 

 more : it treats geometry as an exact science, while it avowedly states 

 that an assumption may be made which is demonstrably incorrect. 



We do not deny that a mind well versed in the doctrine of limits 

 would have a process of its own, by which it would rigorise the methoc 

 of Lacroix, arguing as follows : The same means which show the pro- 

 positions of geometry to have no error perceptible to our senses 

 would also show them to have no error perceptible to any imperfecl 

 senses, however near to perfection they might be; and that error 

 which is less than any assignable error, must be no error at all. All 

 this, wheu properly extended, we might admit ; but how are we to 

 suppose, in the student who has just left the fourth book of Euclid, a 

 perception of the truths of the doctrine of limits, the notorious want 

 of which creates the difficulties of the differential calculus ? Sound 

 teaching makes a true theory of proportion one, out of many previous 

 helps, to the attainment of the differential calculus : but the inversion 

 of the process not only adds difficulty to the latter, by intercepting 

 proper illustration, but introduces falsehood into the former ; and such 

 teaching is the most vicious of all vicious circles, because it propagates 

 its kind. 



The mathematical writers of thig country have, taken altogether, 

 shown a superiority in exactness of demonstration over those of any 

 other of modern times, and the deep and early root which the sound 

 principles of Euclid have taken has been mainly the cause of this. If 

 those principles be abandoned, that superiority will cease to exist ; but 

 this of itself would be of little consequence ; not so the loss of a large 

 amount of formation of accurate habits, which would certainly follow 

 the substitution of a gross and (so called) practical instrument of cal- 

 culation for an exact and liberal science. 



Those to whom the mathematical sciuuces are taught as aids to the 

 power of distinguishing truth from falsehood, logic from fallacy, the 

 exact consequence from all incorrect inference, very many times exceed 

 in number those who only wish for an instrument to be used in the 

 study of physics and the arts of life. If then practical mathematics 

 mean those mathematics which best answer the purpose of the great 

 majority of learners, the more they are rigorous the more they are 

 practical. But the enticing word practical has been otherwise appro- 

 priated ; and " exact enough for practical purposes " is the phrase 

 applied to many a result of which the practical ii.se belongs to the 

 astronomer, mechanician, surveyor, engineer, or computer. To such a 

 meaning of the word practical there is no objection as opposed to 

 liberal or disciplinatory, when it is knowledge or science which is spoken 

 of ; but as applied to art, practical is opposed to unpractical, which 

 cannot be carried into practice, or useless, for that which cannot be 

 practised is useless in art. But those who would consider the use of 

 knowledge in steadying the mind and making it a judge between the 

 true and the false, and a safe guide to the methods of finding truth, 

 must beware, in mathematics, how they allow the notions which art 

 attaches to the word practical any influence over their method of 

 studying science. For want of such a caution many have missed all 

 comprehension of the higher branches even of the art to which they 

 aspired, to say nothing of the loss of that science to which it should 

 seem they did not mean to aspire. 



PROPORTION, in Music, is either Harmonical or Rhythmical. 

 Harmonical Proportion is when, of three numbers representing the 

 relations of sounds, the first has the same proportion to the third as 

 the difference between the first and second has to the difference be- 

 tween the second and third ; as in the numbers 6, 8, 12 ; where 



that is to say, 



12 

 12 



8-6 : 

 2 : 4. 



12-8; 



When four numbers are in Harmonical Proportion, then the first 

 has the same proportion to the fourth as the difference between the 

 first and second has to the difference between the third and fourth : aa 

 in the numbers 6, 8, 12, 18 ; where 



13 

 18 



8-6 : 

 2 : 6. 



18-12; 



that is to say, 



The proportions of the sounds of the diatonic scale [DIATONIC] are 

 as follows : 



The Key-note 

 2nd 

 3rd (major) . 



4th 

 5th 



3 

 5 



15 

 2 



- 

 oth, or octave . 



[SCALE; ACOUSTICS.] 



Rhythmical Proportion is the proportion, in relation to time or mea- 

 sure, between the notes representing duration. Thus, the semibreve 

 to the mimm is 2 : 1 ; the semibreve to the crotchet 4 1 the 

 minim to the semiquaver, 8 : 1 ; &c. That is, the semibreve is twice 

 as long in time as the minim ; four times as long as the crotchet &c 

 PROPORTIONAL COMPASSES. [COMPASSES 1 ' 



PROPORTIONAL LOGARITHMS, also called 'logistic logarithms 

 [lABLES.] Suppose it frequently required to calculate the fourth 

 term of a proportion of which the first term is one given quantity 

 say A : that is, required a fourth proportional to A, p, and q. Common 

 logarithmic calculation here requires three inspections of the table, one 

 addition, and one subtraction. But if A be always the same thing, a 

 new table may be framed, which shall only require two inspections 

 and one addition, as follows : Opposite to p in the table, write log A 

 log p instead of log p, and call the former the proportional loga- 

 rithm of p, which must be considered as the abbreviation of " logarithm 

 of p proper to be used in proportions of which the first term is A." 

 The rule then is ; to find a fourth proportional to A, p, and q add the 

 proportional log of p to that of q, and the sum is the proportional 

 logarithm of the answer. For log A logp, and log A - log a, added 

 together, give 



log A-log9. 



A ' 



which is, by definition, the proportional logarithm of p q -=- A, the 

 answer required. 



In tables made to be used with the old Nautical Almanac, in which 

 the moon's motion was given for every three hours, A was made = 3 h 

 = 10800'; and p and 9 were given in the table, not in seconds, but 

 reduced to hours, minutes, and seconds. Thus the question 



3 b : l k 23 18' : : 14 13- : x, 



could be answered, and x found, by two inspections and an addition. 

 But the convenience of this table lay much more in the arrangement 

 into hours, minutes, and seconds, than in the nature of the substitute 

 for the logarithm : and since a similar arrangement is now made to 

 accompany common tables of logarithms, it may be doubted whether 

 the day of logistic logarithms be not past. 



PROPORTIONAL PARTS, a name given in logarithmic and other 

 tables to small tables which are annexed to the differences of the 

 abular number, and which consist merely in setting down the several 

 tenths of the differences or the nearest whole numbers to them. 

 Thus, in the case of 953, the table of proportional parts is as follows : 



953 



Thus, 286 is the whole number nearest to 3-tenths of 953; from 

 vhich we infer that 29 is the whole number nearest to 3-hundredths 

 >f 953. If then we would have 74 of 953 to the nearest whole 

 number, we take 



7 tenths . 667 



4 huudredths . 38 



705 



>r 705 is the nearest whole number required, subject to the possi- 

 bility of an error of a unit, which is of no consequence in the matters 

 or which such tables are used. This is the process required in 

 ogarithmic interpolation, when tables of seven decimal places are 

 used. Makers of tables now begin to give complete multiples, with 

 ne figure separated, as by a comma. 



PROPORTIONS, DEFINITE. [ATOMIC THEORY 1 



PROPOSITION. [OBOANON.] 



PROPYL (C H 7 ), sometimes called trityltrom rphos, third is 

 he third organic radical in the series C.H.+,. It is the assumed root 

 f the members of the propylic, or tritylic, group of organic com- 

 TOunds. The following are the only members of the propylic group 

 t present known. 



Propylene (C 8 H ). Tritylene. This gas is one of the products of the 

 decomposition of araylic alcohol on passing through a red hot porcelain 

 ube. It is most readily prepared by distilling one part of iodopropy- 



