[§ 118.] In expressing the intermediate numbers, the most common practice is to place the smaller number before the greater one with the conjunction et, or to make the greater number precede the smaller one without et, as quartus et vicesimus, or vicesimus quartus. But there are many instances in in which the smaller number precedes without et ; e. g. quintus tricesimus; and from 13 to 19 this is the ordinary method, though we also find tertius et decimus, decimus tertius, and decimus et tertius. (See Cic. de Invent., i. 53. and 54.) Instead of primus et vicesimus, &c., we find still more frequently unus et vicesimus, fem. una et vicesima, or with the elision of the vowel, unetvicesima, with the genitive unetvicesimae, as in Tacit. Annal. i. 45., and Hist. i. 67. The 22d, 32d, &c., is more frequently and better expressed by alter et vicesimus or vicesimus et alter, than by secundus et vicesimus. Now and then we meet with duoetvicesimus, duoettricesimus, in which case the word duo is indeclinable. The 28th, 38th, &c., are expressed also by duodetricesimus, duodequadragesimus, and the 29th, 39th, 99th, by undetricesimus, undequadragesimus, undecentesimus, the words duo and unus (un) being indeclinable; and both forms are of more. frequent occurrence than octavus and nonus et vicesimus, or vicesimus octavus, vicesimus nonus. There is a class of adjectives in anus which are derived from ordinal numerals, e. g. primanus, secundanus, tertianus, vicesimanus: they express the class or division to which a person belongs; in Roman writers they chiefly denote the legion of the soldiers, whence the first word in their compounds is feminine, e. g. tertiadecimani, quartadecimani, tertia et vicesimani, that is, soldiers of the thirteenth, fourteenth, twenty-third legion. In Tacitus we meet with the forms unetvicesimani and duoetvicesimani.

CHAP. XXX.

III. DISTRIBUTIVE NUMERALS.

[$119.] DISTRIBUTIVE numerals denote an equal number distributed among several objects or at different times, and answer to the questions:-"How many apiece?" and, "How many each

time?" (quoteni ?) They are always used in the plural. The English language having no corresponding numerals has recourse to circumlocution.

Examples. Horat. Serm. i. 4. 86. Saepe tribus lectis videas coenare quaternos, to dine four on each couch; Liv. xxx. 30.: Scipio et Hannibal cum singulis interpretibus congressi sunt, each with an interpreter; Cic. in Verr. ii. 49.: pueri senum septenumve denum annorum senatorium nomen nundinati sunt, boys of sixteen or seventeen years each purchased the title of senator; Liv. v. 30. Senatus consultum factum est, ut agri Vejentani septena jugera plebi dividerentur, each plebeian received seven jugera. The passage in Cicero (ad Att. xvi. 8.), Octavius veteranis quingenos denarios dat, has the same meaning as (ad Fam. x. 32.) Antonius denarios quingenos singulis militibus dat; that is, five hundred denarii to each soldier. When the distributive singuli is expressly added, the cardinal numeral is sometimes used; e. g. Cic. in Verr. ii. 55.: singulis censoribus denarii trecenti ad statuam praetoris imperati sunt.

Hence the distributives are applied in multiplication (with adverbial numerals), the same number being taken several times; e. g. non didicit bis bina quot essent; lunae curriculum conficitur integris quater septenis diebus ; Gellius, xx. 7.: Homerus pueros puellasque Niobae bis senos dicit fuisse, Euripides bis septenos, Sappho bis novenos, Bacchylides et Pindarus bis denos; quidam alii scriptores tres fuisse solos dixerunt. Poets in this case sometimes apply the cardinal numerals; e. g. Horace has, bis quinque viri, i. e. decemviri; and in prose we find decies (vicies, tricies) centum milia, although the form decies centena milia, mentioned above (§ 115.), is much more common.

Distributives are further used, instead of cardinals, with words which have no singular; e. g. bini codicilli, bina post Romulum spolia opima (see § 94.) ; and with those substantives the plural of which, though it has a different signification from the singular, yet retains the meaning of a singular, e. g. aedes, castra, litterae, ludi (§ 96.). It must however be observed, that in this case the Romans commonly used uni instead of singuli, and trini instead of terni, since singuli and terni retain their own distributive signification. We therefore say, for example, bina castra uno die cepit; trinae hodie nuptiae celebrantur; quotidie quinas aut senas litteras accipio; for duo castra would mean "two castles," duae aedes "two temples," and duae litterae "two letters of the alphabet." This, however, is not the case with liberi (children), for this word has not the meaning of a singular (liberi are children, and not a child), and we accordingly say duo liberi, jus trium liberum, &c.

Bini is used for duo to denote things which exist in pairs, as bini boves, binae aures; and in Virgil, Aen. i. 317.: bina manu crispans hastilia. No prose writer goes beyond this in the use of the distributives instead of the cardinals (except in combination with milia, see § 116.). Poets and Pliny the elder use these numerals in the singular in the sense of multiplicatives, e. g. Lucan, viii. 455. septeno gurgite, with a sevenfold whirl; Plin. xvii. 3.: campus fertilis centena quinquagena fruge, with one hundred and fifty fold corn. In the ordinary language they occur only in the plural, and as adjectives of three terminations, i, ae, a.

A longer form of the hundreds: ducenteni, trecenteni, quadringenteni, &c., which is mentioned by Priscian, cannot be proved to exist. Here too there is some freedom in the combination of the numerals: instead of viceni quaterni, we may say quaterni et viceni or quaterni viceni, and for 18 and 19 we have also the forms duodeviceni and undeviceni. The genitive of these numerals is commonly in um instead of orum, as binum, ternum, quaternum, quinum, &c., but not singulum for singulo

rum.

"A thousand each time" might, according to analogy, be expressed by milleni, and then continued bis milleni, ter milleni, &c.; but this form is not in use, and instead of it we say singula milia, bina, terna, quaterna, quina milia; e. g. Sueton. Octav. extr.: Legavit Augustus praetorianis militibus singula milia nummum (that is, one thousand to each), cohortibus urbanis quingenos, legionariis trecenos nummos; Livy: in singulis legionibus Romanis quina milia peditum, treceni equites erant. Milia alone is frequently used for singula milia, if its distributive meaning is indicated by some other word; e.g. Livy, xxxvii. 45.: dabitis milia talentum per duodecim annos, i. e. one thousand talents each year; Curtius, v. 19. : singulis vestrum milia denarium dari jussi, where mille is an incorrect reading; comp. Liv. xxii. 36. This use of the plural, which occurs in other words also, as asses, librae, jugera, with the ellipsis of singuli, ae, a, has been established by J. Fr. Gronovius on Livy, iv. 15. and xxix. 15.; and by Bentley on Horace, Serm. ii. 3. 156.

From these distributives are derived adjectives in arius, which indicate of how many units or equal parts a thing consists, whence they are termed partiaria, e. g. numerus binarius, a number consisting of two units, i. e. two; scrobes ternarii, holes of three feet; versus senarius, a verse of six feet; nummus denarius, a coin of ten units, that is, asses; senex octogenarius, an old man of eighty; rosa centenaria, a rose with one hundred leaves; cohors quingenaria, of 500 men. The word numerus is most frequently combined with these adjectives, to supply the

place of the substantives unio, binio, ternio, which are not based on very good authority. (See § 75.) Singularis and milliarius are more commonly used instead of singularius, millenarius.

CHAP. XXXI.

IV. MULTIPLICATIVE NUMERALS.

[§ 120.] MULTIPLICATIVES answer to the question, "How many fold?" (quotuplex ?) They are: simplex, duplex, triplex, quadruplex, quincuplex, septemplex, decemplex, centuplex. These are the only ones that can be shown to have been in use. Sixfold does not occur in Latin; it might be sexuplex or seplex, but not sextuplex, as some grammarians assert. Octuplex is attested by the derivative octuplicatus, and novemplex by the analogy of septemplex. (Modern writers use also: undecimplex, duodecimplex, sedecimplex, vicecuplex, tricecuplex, quadragecuplex, quinquagecuplex, sexagecuplex, septuagecuplex, octogecuplex, nonagecuplex, ducentuplex, trecentuplex, quadringentuplex, quingentuplex, octingentuplex, &c., and millecuplex.)

It will not be out of place here to add the Latin expressions for fractions, which are always denoted by parsis dimidia pars, tertia pars, quarta pars, quinta, sexta, septima pars, &c. In cases where the number of the parts into which a thing is divided, exceeds the number of parts mentioned only by one, as in 3, 3, 3, the fractions are expressed in Latin simply by duae, tres, quattuor partes, that is, two out of three, three out of four, and four out of five parts: may be expressed by octava pars, or by dimidia quarta. In all other cases fractions are expressed as in English: , duae septimae ; 4, tres septimae, &c., or the fraction is broken up into its parts, e. g. by pars dimidia (3) et tertia (); and 10 by tertia et septima.

CHAP. XXXII.

V. PROPORTIONAL NUMERALS.

[§ 121.] PROPORTIONAL numerals express how many times more one thing is than another, but they cannot be used throughout. They answer to the question quotuplus? They are: simplus, a, um; duplus, triplus, quadruplus, quinquiplus, (probably sexuplus,) septuplus, octuplus, (perhaps nonuplus,) decuplus, centuplus, and according to the same analogy we might form ducentuplus, and so on, as in the multiplicatives above. But they are almost universally found only in the neuter.

CHAP. XXXIII.

VI. NUMERAL ADVERBS.

[§ 122.] 1. THE numeral adverbs answer to the question, " How many times?" quotiens? to which totiens is the demonstrative, and aliquotiens the indefinite. The form in ns is the original, and prevailed in the best periods of the language; subsequently the termination es was preferred in numerals, but ens still remained in the words just mentioned.