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agreed to go. As the journey was not a long or dangerous one, the servants of Balak returned at once to inform their master of their success, leaving Balaam to follow at his own convenience. So Balaam, still without consulting Yahweh, saddled his ass and set out for Moab, attended only by two servants. The land through which he had to pass, so far from being a desert, was a land of oil and wine; and when Balaam was riding along a narrow path between two vineyards, the angel of Yahweh would have slain him, had not his ass swerved and saved him. That this episode belongs to J no one need ever forget, since the only parallel in Scripture to the speaking ass is the serpent that spoke in Eden. Balaam, after being sternly rebuked, was allowed to proceed, but only on condition that “the word that I shall speak to thee, that thou shalt speak.” Balak met Balaam at Ar-Moab, whence they went to Kiriath-Huzoth and thence to the top of Peor. There Balaam blessed Israel. Balak angrily taunted Balaam with having lost the honours intended for him, and bade him flee to his own place. Balaam reminded Balak of his declaration that he could not go beyond the word of Yahweh, and then boldly announced the respective destinies of Israel and Moab, xxiv. 15-19.

As seven is the perfect number and as Balaam had ordered seven altars to be built, the Redactor thought it would be well to have seven Mĕshālîm or metrical oracles; and so he added other three which are certainly not pertinent to the situation, as they allude not merely to the Assyrian empire but to the Macedonian, and even, as some maintain, to the Roman empire, cf. xxiv. 24.

The poetical quotations in Numbers are of the utmost importance, not only as helping to determine the date of the book but as indicating the value of poetry in its bearing on history. In xxi. 14 we have a poetical quotation from a lost volume of early poetry entitled “The Book of the Wars of Yahweh.” It is highly probable that Deborah’s song was also originally in this book; and when we compare the statement in that song as to Israel’s full fighting strength, viz. 40,000 men, with the statements in the prose of Numbers as to 600,000 men and more, we at once realise how much closer to actual facts we are brought by early poetry than by the later prose of writers like P. Perhaps it is in chap. xxxi. that we have the clearest proof of the non-historical character of the book. There we are told that 12,000 Israelites, without losing a single man, slew every male Midianite, children included, and every Midianite woman that had known a man, and took so much booty that there had to be special legislation as to how is should be divided. But if this were actual fact, how could the Midianites have ever reappeared in history? And yet in Gideon’s time they were strong enough to oppress Israel. From this chapter, unhistorical as it must be, we see how the legislation of Israel, whatever its character or origin, was referred back to Moses the great Law giver of Israel.  (J. A. P.*) 

NUMBERS, PARTITION OF. This mathematical subject, created by Euler, though relating essentially to positive integer numbers, is scarcely regarded as a part of the Theory of Numbers (see Number). We consider in it a number as made up by the addition of other numbers: thus the partitions of the successive numbers 1, 2, 3, 4, 5, 6, &c., are as follows:—

1;

2, 11;

3, 21, 111;

4, 31, 22, 211, 1111

5, 41, 32, 311, 221, 2111, 11111;

6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.

These are formed each from the preceding ones; thus, to form the partitions of 6 we take first 6; secondly, 5 prefixed to each of the partitions of 1 (that is, 51); thirdly, 4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to each of the partitions of 3 (that is, 33, 321, 3111); fifthly, 2 prefixed, not to each of the partitions of 4, but only to those partitions which begin with a number not exceeding 2 (that is, 222, 2211, 21111); and lastly, 1 prefixed to all the partitions of 5 which begin with a number not exceeding 1 (that is, 111111); and so in other cases.

The method gives all the partitions of a number, but we may consider different classes of partitions: the partitions into a given number of parts, or into not more than a given number of parts; or the partitions into given parts, either with repetitions or without repetitions, &c. It is possible, for any particular class of partitions, to obtain methods more or less easy for the formation of the partitions either of a given number or of the successive numbers 1, 2, 3, &c. And of course in any case, having obtained the partitions, we can count them and so obtain the number of partitions.

Another method is by L. F. A. Arbogast’s rule of the last and the last but one; in fact, taking the value of 𝑎 to be unity, and, understanding this letter in each term, the rule gives 𝑏; 𝑐, 𝑏²; 𝑑, 𝑏𝑐, 𝑏³; 𝑒, 𝑏𝑑, 𝑐², 𝑏²𝑐, 𝑏⁴, &c., which, if 𝑏, 𝑐, 𝑑, 𝑒, &c., denote 1, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively.

An important notion is that of conjugate partitions. Thus a partition of 6 is 42; writing this in the form ${\Big \{}$1111,
11 and summing the columns instead of the lines, we obtain the conjugate partition 2211; evidently, starting from 2211, the conjugate partition is 42. If we form all the partitions of 6 into not more than three parts, these are

	6,	51,	42,	33,	411,	321,	222,
and the conjugates are
	111111,	21111,	2211,	222,	3111,	321,	33,

where no part is greater than 3; and so in general we have the theorem, the number of partitions of 𝑛 into not more than 𝑘 parts is equal to the number of partitions of 𝑛 with no part greater than 𝑘.

We have for the number of partitions an analytical theory depending on generating functions; thus for the partitions of a number n with the parts 1, 2, 3, 4, 5, &c., without repetitions, writing down the product

1+𝑥.1+𝑥²,1+𝑥³.1+𝑥⁴. . . , ＝1+𝑥+𝑥²+2𝑥³. . .+N𝑥^𝑛+. . . ,

it is clear that, if 𝑥^α, 𝑥^β, 𝑥^γ, . . . are terms of the series 𝑥, 𝑥², 𝑥³, . . . for which α+β+γ+ . . =𝑛, then we have in the development of the product a term 𝑥^𝑛, and hence that in the term N𝑥^𝑛 of the product the coefficient N is equal to the number of partitions of 𝑛 with the parts 1, 2, 3, . . . , without repetitions; or say that the product is the generating function (G. F.) for the number of such partitions. And so in other cases we obtain a generating function.

Thus for the function

1/1−𝑥.1−𝑥².1−𝑥³. . .,＝1+𝑥+2𝑥²+ . . +N𝑥^𝑛+. . . ,

observing that any factor 1/1-𝑥^𝑙 is=1+𝑥^𝑙+𝑥^2𝑙+. . , we see that in the term N𝑥^𝑛 the coefficient is equal to the number of partitions of 𝑛, with the parts 1, 2, 3, . . , with repetitions.

Introducing another letter 𝑧, and considering the function

1+𝑥𝑧.1+𝑥²𝑧. . .,＝1+𝑧(𝑥+𝑥²+. .). . . +N𝑥^𝑛𝑧^𝑘+. . ,

we see that in the term N𝑥^𝑛𝑧^𝑘 of the development the coefficient N is equal to the number of partitions of 𝑛 into 𝑘 parts, with the parts 1, 2, 3, 4, . . ., without repetitions.

And similarly, considering the function

1/1−𝑥𝑧.1−𝑥²𝑧.1−𝑥³𝑧. .,＝1+𝑧(𝑥+𝑥²+ . .) . . . +N𝑥^𝑛𝑧^𝑘+. .

we see that in the term N𝑥^𝑛𝑧^𝑘 of the development the coefficient N is equal to the number of partitions of 𝑛 into 𝑘 parts, with the parts 1, 2, 3, 4, . . . , with repetitions.

We have such analytical formulae as

1/1−𝑥𝑧.1−𝑥²𝑧.1−𝑥³𝑧. .,＝ 1+𝑧𝑥/1−𝑥+𝑧²𝑥²/1−𝑥 . 1−𝑥+. . . ,

which lead to theorems in the partition of numbers. A remarkable theorem is

1−𝑥.1−𝑥².1 −𝑥³. 1−𝑥⁴. = 1−𝑥−𝑥²+𝑥⁵+𝑥⁷−𝑥¹²−𝑥¹⁵+. . . ,

where the only terms are those with an exponent 1/2(3𝑛²±𝑛), and for each such pair of terms the coefficient is (−)^𝑛1 The formula shows that except for numbers of the form 1/2(3𝑛²±𝑛) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity. Thus for the number 11, which is not an excepted number, the two sets of partitions are

11,	821,	731,	641,	632,	542
10.1,	92,	83,	74,	65.	5321,

in each set 6.

We have

1−𝑥.1+𝑥.1+𝑥².1+𝑥⁴.1+𝑥⁸...＝1;

or, as this may be written,

1+𝑥.1+𝑥².1+𝑥⁴.1+𝑥⁸. . .＝1/1−𝑥, ＝1+𝑥+𝑥²+𝑥³+. . . ,

showing that a number 𝑛 can always be made up, and in one way only, with the parts 1, 2, 4, 8, . . . The product on the left-hand side may be taken to 𝑘 terms only, thus if 𝑘=4, we have

1+𝑥.1+𝑥².1+𝑥⁴.1+𝑥⁸,＝1−𝑥¹⁶/1−𝑥,＝1+𝑥+𝑥². . . +𝑥¹⁵,