On account of this difficulty, the atomic weights published by Dalton, and the more accurate ones of Berzelius, were not always identical with the values now accepted, but were often simple multiples or submultiples of these.
Each of the four faces of each rod contains multiples of one of the nine digits, and is similar to one of the slips just described, the first rod containing the multiples of o, I, 9, 8, the second of o, 2, 9, 7, the third of o, 3, 9, 6, the fourth of 0, 4, 9, 5, the fifth of I, 2, 8, 7, the sixth of I, 3, 8, 6, the seventh of I, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5.
Each rod therefore contains on two of its faces multiples of digits which are complementary to those on the other two faces; and the multiples of a digit and of its complement are reversed in position.
The masses of different elements which combine separately with one and the same mass of another element, are either the same as, or simple multiples of, the masses of these different elements which combine with each other.
Then a binary n", equated to zero, represents n straight lines through the origin, and the x, y of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates.
As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.
If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.
Notation of Multiples.-The above is arithmetic. The only thing which it is necessary to import from algebra is the notation by which we write 2X instead of 2 X X or 2.
Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.
Thus, the gases are not present in simple multiples of their combining weights; atmospheric air results when oxygen and nitrogen are mixed in the prescribed ratio, the mixing being unattended by any manifestation of energy, such as is invariably associated with a chemical action; the gases may be mechanically separated by atmolysis, i.e.
(c) the standard gallon (and multiples and fractions of it), declared to contain 10 lb of water at 62° F., being in volume 277.274 cub.
Troy weight was abolished, from the 1st of January 1879, by the Weights and Measures Act 1878, with the exception only of the Troy ounce, its decimal parts and multiples, legalized in 1853, 16 Vict.
Again, there are many theories of the equivalence of different cubic cubits of water with various multiples of talents (2, 3, 18, 24, 33); but connexion by lesser units would be far more probable, as the primary use of weights is not to weigh large cubical vessels of liquid, but rather small portions of precious metals.
The multiples of the 20.63 cubit are in late times generally reckoned in these feet of 2/3 cubit.
In these cases the digit itself, or decimal multiples, seem to have been used.
± 0.6; by 9 unmarked pottery measures (30) 29.1 ± 0.16, and divided by 20; by 18 vases, supposed multiples of hon (1), 32.1 ± 0.2.
From ratios stated before Greek times (35) the series of multiples was --
The system of multiples was for liquids --
From the east this unit passed to Asia Minor; and six multiples of 2 to 20 shekels (av.
It has been connected theoretically with a binary division of the 10 shekels or "stone" of the Assyrian systems (28), 1290/16 being 80.625; this is suggested by the most usual multiples being 40 and 80 = 25 and 50 shekels of 129; it is thus akin to the mina of 50 shekels previously noticed.
A beautiful set of multiples of the scripulum was found near Lyons (38), from 1 to 10 x 17.28 grains, showing a libra of 4976.
In this table multiples of the differences, instead of proportional parts, are given.'
In 1815 he published anonymously in the Annals of Philosophy a paper "On the relation between the specific gravities of bodies in their gaseous state and the weights of their atoms," in which he calculated that the atomic weights of a number of the elements are multiples of that of hydrogen; and in a second paper published in the same periodical the following year he suggested that the rrpcbrn iiXrl of the ancients is realized in hydrogen, from which the other elements are formed by some process of condensation or grouping.
(a) Adding multiples of a fraction to produce a more convenient fraction (perhaps connected with the use of palms and cubits in decoration in a proportion based on the number 8).
Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth.
Then, if possible, B and, C themselves are to be resolved each into rnI factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.
An illiterate population is very prone to state its age in even multiples of five, and even where education is widely spread this tendency is not altogether absent, as may be seen from the examples given in TABLE III.
The selection of three as their number has been supposed to refer to the most ancient division of the year into spring, summer and winter, but it is probably only another instance of the Greek liking for that particular number or its multiples in such connexions (three Moerae, Charites, Gorgons, nine Muses).
2 Towards the end of the third quarter of the 19th century it therefore became clear that electricity, whatever be its nature, was associated with atoms of matter in the form of exact multiples of an indivisible minimum electric charge which may be considered to be " Nature's unit of electricity."
The values of coin and notes are expressed in multiples of the real (plural reis), a monetary unit which does not actually exist.
3.3 (iii.) Multiples, Submultiples and Quotients
I, V, X, L, C, D, M, proceeding by multiples of five and two alternately.
The Hebrews had a notation containing separate signs (the letters of the alphabet) for numbers from t to to, then for multiplies of to up to zoo, and then for multiples of too up to 400, and later up to moo.
(iii) The multiples of to are usually strongly marked; but special stress is also laid on other important numbers, e.g.
The multiples of 12.
Are the successive multiples of 5d.
We find the multiples of 12d.
Multiple-Tables.-The diagram C or D of § 35 is part of a complete table giving the successive multiples of the particular unit.
If we take several different units, and write down their successive multiples in parallel columns, preceded by the numberseries, we obtain a multiple-table such as the following: It is to be considered that each column may extend downwards indefinitely.
Submultiples.-The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive submultiples being one-half, one-third, one-fourth,..
In §§ 33 4 2 We Have Dealt With The Parallelism Of The Original Number Series With A Series Consisting Of The Corresponding Multiples Of Some Unit, Whether A Number Or A Numerical Quantity; And The Relations Arising Out Of Multiplication, Division, &C., Have Been Exhibited By Diagrams Comprising Pairs Of Corresponding Terms Of The Two Series.
We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations.