C O M 



22 



COM 



be 3, 10, 11, 18, &c. of such parts; the two series dif- 

 fering 2, or the major diesis L — /, in each case ; for it 

 will be observed, that the wolf always arises, by the 

 substitution of L for /, or vice versa. And all such 

 false intervals, called wolves, differ a major diesis (d) 

 from the regularly tempered intervals of the same name 

 or degree. 



In commensurable systems, if the octave and any one 

 other of the 22 intervals in the Tables, or any two 

 among them be given, it will be easy to find the va- 

 lues of L and / therefrom, and thence the values of all 

 the remainder of the 23 intervals : thus, if the octave be 

 given = iO, (of any certain intervals or parts,) and the 

 fifth = 29 (of the same parts,) then 7 L + 5 I = 50, 

 and 4 L -f 3 I = 29, whence 3 L -f- 2 / = 21, and 

 L + / = 8 ; L = 8 — /, and 24 — 3 I + 2 / = 21, or 

 3 = 1, and 5 = L ; whence every thing else in the 

 system may readily be found. So also, if the tempered 

 III were given = 1 6 parts, and the Vlth wolf— 39 

 parts, we have 2 L + 2 / = 16, or L + I = 8, and 

 6 L + 3 I = 39, or 2 L + I = 13 ; whence will be 

 found L = 5 and / = 3, as before. 



Merely giving the number of equal parts, into which 

 the octave of a commensurate system is divided, is in- 

 sufficient to determine that system, unless it be further 

 given or assumed, that T and L, (and consequently L 

 and /) are whole numbers, as Mr Farey has shewn in the 

 Philosophical Magazine, vol. xxxviii. p. 435, and has 

 there subjoined a general rule for finding all the possi- 

 ble and practicable answers to such questions, which 

 rule (for T and L) we shall not repeat here, but give a 

 similar one adapted to L and /, which will be found 

 more useful, in conjunction with our Tables above. In 

 the case of the octave divided into 50 equal parts, as 

 before mentioned, all the possible integer values of L 

 and / are as follows : viz. 



L I 



7X0 + 5X10 = 50 

 7 X 5 + 5 x 3 = 50 

 And wherein it is plain that only the second line is ap- 

 plicable to a system of intervals, and gives L = 5 and 

 /. — 3, as before found. 



Our general rule is as follows: viz. From the given Commen- 

 number of equal parts (a) in the octave, deduct succes- surable 

 sively the multiples of 7, in the series 0, 7, 14, 21, 28, s y stem »- 

 35, &c. until a remainder is found divisible by 5, or X *"" "Y™"""' 

 which ends in or 5, as all such numbers do, and let 

 such multiple of 7, or subtrahend, be called b ; then will 



— — be the greatest value of I, and —the least, or cor- 



J 7 



responding value of L ; and all other corresponding 

 values of / will decrease from this by 7, and those L in- 

 crease by 5, as in the example above. 



Since L and / both represent half notes, or tones, it 

 is plain that they can never, in practicable systems, dif. 

 fer much from each other, yet they never will coincide j 

 except in the equal temperament, or isotonic system, 

 and it will always be easy to select that line among the 

 possible values of L and /, which answers the intended 

 conditions. As another example, suppose the octave 

 divided into 612 equal parts, or artificial commas of 

 Farey, then 



L / 



7X 1+5X121=612 



7 X 6 + 5 x 114=612 



7 X 11 +5 x 107=612 



7 X 16 + 5 X 100 = 612 



7 X 21 +5 X 93 = 612 



7 X 26 + 5 X 86 = 612 



7 X 31 +5 X 79 = 612 



7 X 36 + 5 X 72 = 612 



7 X 41 + 5 x 65 = 612 



7 X 46 + 5 x 58 = 612 



7 X 51 +5 X 51 = 612 



7 X 56+ 5 x 44 = 612 

 &c. &c. 



shew the possible values of L and I : the last line but 

 one shewing that 51 might answer for both L and /, 

 each of which are, in such cases, the 12th part of the 

 octave, for the isotonic or equal temperament system. 

 The last line gives L = 56 and I = 44, which are in 

 the ratio of 14 and 11, and answer also to 153 parts in 

 the octave, &c. (^) 



COMMERCE. 



^^X^' ^ 0HMERCE -is frequently used in the limited sense of 

 " mercantile intercourse with a foreign country," while 

 trade is made to denote " traffic among the inhabitants of 

 the same country." Strictly speaking, however, there is 

 no foundation for this distinction, either word being 

 entitled, both by etymology and decisive authority, to a 

 comprehensive and general interpretation. Either may 

 therefore be defined, in the broadest sense, " an ex- 

 change of commodities, whether the object of the ex- 

 change be consumption or re-sale at a profit." We 

 ^hall divide our discussion of tins subject into three 

 heads. 



1. Historical sketch of commerce. 

 -2. Present state of British commerce. 

 3. Observations on the principles of commerce. 



E 1 i->rory of I. Historical sketch of commerce. — The origin of corn- 

 Commerce, merge is involved in the same obscurity as the other oc- 

 cupations of an early stage of society. Its beginnings, 

 like those of other things, must have been extremely 

 rude, and must have consisted in nothing more than a 



barter of one simple article for another. In progress of Commerce, 

 time, certain articles of general utility, and which were "■■ "V""' 

 capable of transportation, such as brass, iron, cattle, &c. 

 became a kind of standard for the purchase of other 

 commodities. Such appears to have been the case at the 

 time of the Trojan war, and likewise, probably at the 

 later period, when Homer composed his poem. A far- 

 ther improvement in the mode of managing commerce, 

 consisted in the adoption of gold and silver as money, 

 or general representatives of value. This important 

 point once established, it was easy to make the addi- 

 tional progress of stamping on the metal an official mark 

 declaratory of its weight ; and eventually, to constitute 

 the mark, in the shape of coin, an indication of its fine- 

 ness as well as of its weight. An attention to the dif- 

 ferent stages in the progress of currency, affords the in- 

 quirer a good criterion of the advance of commerce. In 

 maritime countries a similar rule is offered by the pro- 

 gress of navigation. The conveyance of merchandize 

 by water is, after mankind emerge from primitive ig- 

 norance, so much easier than by land, that we may safe- 



