his experiment with 5760 grs. of muriatic acid and 2393 grs. of chalk.

1000

"If now we desire to find the proportion of the elements in the pure salt forming a neutral body, we must first seek to determine the amount of lime out of the weight of the crude lime used or the aerial salt of lime. This amounts to 2393 grains. According, then, to par. 1, 1000:559=2393 : lime, and the lime is equal to 230551337; this, when subtracted from the 2544 grains of the neutral mass obtained, leaves a residue of 1207 grains, the weight of the muriatic acid. If, then, 1207:2544*=1000: 1107, it is clear that in the salt of lime (chloride of calcium) 1000 parts of muriatic acid are united in a neutral state with 1107 parts of lime; the proportion of the elements in this neutral solution is then best designated by 1000:1107.

In this manner all the earths are treated, after which he gives the relation of the quantities of alkaline earths towards sulphuric acid and each other.

Order of the masses of alkaline earths towards muriatic acid. § XXII. P. 27.

"If we set in a row the numbers which have been found representing the masses of alkaline earths which unite with 1000 parts of muriatic acid, we obtain the first series of quantities of the alkaline earths. The muriatic acid is the determining element (elementum determinans) of this series, and every member of this series represents an element determined (elementum determinatum). In order to designate the elements to which we affix these numbers, we shall make use of the chemical signs for the sake of convenience, setting the determining element, or rather its sign, at the top, or at the side of the series of quantities; and when no number is placed, we shall suppose it to be 1000. In order to fix these signs in our memory, we shall here repeat them." (These signs it is not convenient to use.)

*This evidently ought to have been 1337.

66

According to the paragraphs quoted, the following is the series of the alkaline earths in their relation to muriatic

"Little as the members of this series appear to follow any certain order, it is nevertheless decidedly the case; at the same time the inquiry into the law of these series is one of the most difficult problems which stoechiometry gives us to solve, and if we do not go to the inquiry with sufficient practical and theoretical exactness, we shall not succeed in our inquiry into these laws or orders (of the numbers). And now, to inquire into the law of the series before us, let us first seek the difference between each member and its successor, and we obtain 858-734-124, 1107—858-249, 3099-1107-1992. Let us then use the first difference to divide the two following differences, and we obtain !=2+1,

8

1992

=1611. Then let us see if one quotient allows of division by the other, that is, let us divide 16+ by 2+. If we bring divisor and dividend under the same denomination of 124, this will be 16+2 and 2+1=1; therefore 1992:124-1992 and 249 is contained exactly 8 times in 1992, consequently 2=8. From this it is clear, that when we have the first difference 1241-2, all the succeeding differences may be so divided by it that nothing remains; the half here mentioned is only in 858 parts 0.0006 and still less in the other members of the series, it is, therefore, of no importance; it is impossible in experiments to arrive at such minuteness, at the same time in calculating the proportion to 1000 parts, it was necessary to throw away small unimportant fractions, otherwise it would be needful to use an enormous number of figures in order to designate the quantities. Now 232 X 2=249; and 242 X8 1992, consequently 734+22=8581, 734+22+2 X 2=1107; 734

+2+248x2+2x10-3099. In order better to understand all, let us make 734=a, 242=b, then 734-a, 8581=a+b, 11071=a+b+2b=a+3b, 30991=a+b+2b+16b=a+19b. From this the quantitative series appears in the following order :

"Now this series remains always the same, even when we put a higher or a lower number for the mass of the determining element, for if the mass of the determining element is n times greater or n times smaller, then, in the first case, all the terms would be n times greater; that is, multiplied by n; and in the latter case n times smaller or divided by n, and the order of the differences would remain always the same; because what occurs with one of the differences must occur also with the others, if otherwise the determining element must still be considered as such. When this series is attentively considered, we observe that the difference of the successive terms is a mathematical product of the first difference b with an odd number. According to it the quantities in which the hitherto known alkaline earths assert their neutrality with muriatic acid are terms of a real arithmetical progression, the terms of which are found, when the product of a certain quantity with an odd number is added to the first term, only that between them many odd numbers, such as 5, 7, 9, 11, 13, 17, are left out. This is more remarkable, as the differences which the first term makes with the succeeding ones may be represented entirely by odd numbers; for one need only suppose that the mass of the determining element is divided by b, then all the terms of the series would be at once divided by b, and appear in the following form :—

a

"In this case the first term would be and if it were all expressed in numbers then would ===5+38 and the mass of the determining element 1900-19:2249

8. In this way all the members are obtained in numbers, when 1, 3, and 19 are added to the first term 47, and the elements observed which are designated by these figures. It is very probable that the terms +5, +7,8 +9, +11,

+13, +15, +17 are wanting in the series, and the reasons for considering this probable, will be shewn in a suitable place.

"Preliminary determination of the order of the alkaline earths which enter into neutrality with vitriolic acid. § XXIII., p. 33.

"If we put in order the amount (mass) of alkaline earths which stand in neutrality with 1000 parts of vitriolic acid, in the manner adopted with muriatic acid, the following series of quantities is obtained :—

"In order to discover the law of this series let us, as in the former case, subtract the first term from all the succeeding, and we receive 796-616-180; 1053-616-437, 2226-616

77

1809

1610. Let us see now if the first difference can so divide all the rest, that nothing, or at least very little, remains, then · 193=2+15%, 19=8+138. As nothing can be discovered here on account of the variety in the remaining fractions, let us divide every difference by 90 as the half of the first difference, and we obtain =2, 432=5, 1610-18-18. The fractions here are not so considerable as before, although too large to be thrown away. Until, therefore, we are able to complete the order let us make 616–616, 796=616—2.90,* 1053-616+5.90-18, 2226-616+18.90-10"

"Nearer determination of the law by which the quantities of the alkaline earths, which enter into rest and neutrality with muriatic and vitriolic acid, increase or diminish in arithmetical progression. § XXIV., p. 34.

"A. As we cannot completely ascertain the law by which the terms of the two series of numbers obtained by experiment proceed, we must try another source of information, to the obtaining of which the series itself which the determining element of muriatic acid makes with the alkaline earths, gives us an opportunity. As the differences of the quantities in § XXII. are a product of a quantity b with an odd number, it is possible that as many terms are wanting as there are odd numbers between 3 and 19, and even that other terms may lie beyond the term a+196 or +19. Suppose, then, that this series were complete, namely, a, a+b, a+3b, a+5b, a+7b, a+9b, a+11b, a+13b, a+15b, a+17b, a+19b, a+21b, a+236, &c., the masses of the elements which enter into neutrality with 1000 parts of muriatic acid would be the following: