"Now let us suppose that these alkaline earths, which are partly real, partly possible, and designated by the above numbers, entered with muriatic acid into such neutral combinations as would decompose with a neutral compound out of the series § XXIII., such as the magnesia salt, by double affinity either positive or negative (see Pure Stochiometry. Theor. 3, coroll. 3. Introd. definition 16), then only that neutral compound is excepted which the muriatic acid makes with the alkaline element of the neutral salt which we have chosen in the series XXIII., or the combination which is also taken with the magnesia salt. But according to experiment 6, coroll. 2, in the Pure Stochiometry, in the decomposition by double affinity, out of three proportions the fourth may be determined. Let us suppose, then, that all the mentioned actual and possible neutral combinations, magnesia salt excepted, decompose with sulphate of magnesia (bitter salt) either positively or negatively, so that each constituent is placed in a state of rest (Pure Stoech. Theor. 1, coroll. 1); then we may find how many measures of each of the real and possible elements are wanted for 1000 parts of the vitriolic acid measures. (Pure Stoech. Introd. def. 14.)

"B. The first neutral compound in the series, § XXII., is an actual one, namely, alum salt, where 734 parts of alumina stand in neutrality with 1000 parts of muriatic acid. If this neutral or middle salt decomposes with sulphate of magnesia by double affinity, then 858 parts of magnesia must be contained in the sulphate of magnesia, because the proportionate quantity in the magnesia salt is as 1000:858, or rather 1000: 858. § XXII. Now the proportionate quantity in the sulphate of magnesia is 1000: 616 (§ XIX., XXIII.) and 616 1000=858: 1,394; that is, if 616 parts of magnesia insist on rest with 1000 parts of vitriolic acid, the same must occur between 858 parts of the first and 1394 parts of the latter; therefore when alumina salt and magnesia salt decompose, 1000 parts of muriatic acid dissolve with 8581

parts of magnesia, and 1394 parts of vitriolic acid with 734 parts of alumina; the proportionate amount of the alum formed will then be 1394: 734-1000: 526, which is not the proportion of the neutral but the common alum. (§ XXI.) The quantity of the alkaline earth in common alum belongs accordingly to the series § XXIII.

"C. But when, in the decomposition of the first neutral compound, where muriatic acid is the determining element with sulphate of magnesia, the amount of the vitriolic acid is 1394, it is so in all subsequent possible decompositions which the neutral compounds of the other actual and possible elements of the series § XXII. make with the sulphate of magnesia, let these decompositions be positive or negative. (Pure Stoch. Theor. 3, coroll. 3.) The quantity, then, of real and possible elements which belong to 1000 parts of muriatic acid belong to 1394 of vitriolic, and the following proportions are obtained for the compounds where the vitriolic acid becomes at rest with the real and possible elements (Pure Stoech. Theor. 1, coroll. 1), all of which obtained by experiment except the common alum are neutral.

§ XXV.

"A. When we look on all these numbers found, viz., 526, 616, 796, 973, &c., as quantities of the elements which are at rest with 1000 parts of sulphuric acid, we obtain a series, the law of which soon appears to us. Let us first subtract the first term from all the succeeding, and we obtain the following differences, which may be expressed in various ways:

:

"B. The law by which the differences of the actual and possible alkaline earths increase in relation to vitriolic acid is then so far made out that it follows from the product of a number which is here 90 with the consecutive odd numbers; we may consider the numbers which are to be subtracted, such as 3, 4, 6, 8, 9, &c., as nothing, because the greatest error that could be caused is only=1=0.0066, or ‰; but even this is not necessary, because the numbers themselves proceed in distinct order, as the series shews; and if we inquire, as in § XXII., into the manner in which these numbers progress, we observe that if three of them increase

by odd numbers, the succeeding four increase in the ordinary way by one, and so alternately; for example.

"These quantities accordingly proceed in arithmetical progression down to the most insignificant fractions, as we may see by a glance at them.

"The quantities in which the alkaline earths enter into neutrality with muriatic acid are terms of an endless series, which increase by the product of a determinate quantity with the consecutive odd numbers. The same thing occurs with the alkaline earths in relation to sulphuric acid, only that in this case a quantity must be taken from the terms of the last series, the first three excepted; this quantity increasing also in progression.

§ XXVI.

"A. After finding out the law by which the quantities of the alkaline earths increase towards the two acids (sulphuric and muriatic), it becomes necessary to form the series themselves, that we may see clearly the correctness of the proposition advanced as a hypothesis; for if it is done rightly the proposition ceases to be a hypothesis. We shall designate the terms that are wanting in both series by a star, and the

elements which produce with acids a very violent heat when freed from their air, for example, lime and magnesia, by A, as the sign of fire.

"B. Before we set down the quantitative progression in the case of vitriolic acid, we must first inquire if the quantity of alumina in neutral alum belongs to this series; it is 1053. Let us subtract 526 from 1053, and we obtain 527; now 527= 540—13—6.90—13, and consequently 1053=526+6.90—13. But as the series determined by vitriolic acid proceeds by the uninterrupted odd numbers, and no neutral alum can be found in decomposing by double affinity, the quantity 526+6.90—13 does not belong to this series. We must take it, however, in the meantime into the series, because it belongs to the quantities which enter into neutrality. We shall, however, put such in brackets, as must happen when considering the quantity of alumina in common alum, if it is not a legitimate member of the series, and capable of double affinity.