Chapter II: Front Matter (2)

Although the foregoing would be the best division of the musical scale, if our sole object were to render the harmony of its concords as nearly equal as possible, yet the two other considerations, stated at the beginning of the essay, must by no means be neglected, as has been done by Dr. Smith. It seems to be universally admitted, that the sum of the temperaments may be increased to a certain extent, in order to equalize the harmony of the concords; otherwise the natural scale of major and minor tones, which makes the sum of the temperaments of the Vths, IIIds, and 3ds but 2 commas, ought to be left unaltered. Yet how far this principle ought to be carried, may be a matter of doubt. If we make the IIIds perfect, and flatten the Vths and 3ds each ¼ _c_, according to the old system of mean tones, we shall have the smallest aggregate of temperaments which admits of the different concords of the same name being rendered equally imperfect; but this amounts to 2½ commas. Thus far, however, it seems evidently proper to proceed. If we go still farther, and endeavour to equalize the harmony of the concords of _different_ names, it may be questioned whether nearly as much is not lost as gained; for the aggregate temperaments are increased, in Dr. Smith's scale, to 2⅔ _c_, and in that of the above proposition to 2-5/7 _c_. The system of mean tones, although more unequal in its harmony when but two notes are struck at once, yet when the chords are played full, as they generally are on the organ, never offends the ear by a transition from a better to a worse harmony. For every _triad_ is equally harmonious; being composed of a perfect IIId, and a Vth and 3d, tempered each ¼ _c_, or of their complements to, or compounds with octaves, which, in their kinds, are equally harmonious.

Again, if different chords, in practice, vary in the frequency of their occurrence, this will be a sufficient reason for deviating from the system of equal temperament. Suppose, for example, that a given sum of temperament is to be divided between two Vths, one of which occurs in playing ten times as often as the other: there can be no doubt that the greater part of the temperament ought to be thrown upon the latter. Hence it becomes an important problem to ascertain, with some degree of precision, the relative frequency with which different consonances occur in practice. Before proceeding to a direct investigation of this problem, it may be observed, in general, that such a difference manifestly exists. In a given key, it cannot have escaped the most superficial observer, that the most frequent combination of sounds is the common chord on the tonic; that the next after this is that on the dominant, and the third, that on the subdominant. Perhaps scarcely a piece of music can be found, in which this order of frequency does not hold true. It is equally true that some signatures occur oftener than others. That of one sharp will be found to be more used, in the major mode, than any other; and, in general, the more simple keys will be found of more frequent occurrence than those which have more flats or sharps. These differences are not the result of accident. The tonic, dominant, and subdominant, are obviously the most prominent notes in the scale, and must always be the fundamental bases of more chords than either of the others; while the greater ease of playing on the simpler keys will always be a reason with composers for setting a larger part of their music on these, than on the more difficult keys. It is observable, that the greater part of musical compositions, whether of the major or minor mode, is reducible to two kinds: that in which the base chiefly moves between the tonic and its octave, and that in which the base moves between the dominant and subdominant of the key. The former class, in the major mode, are almost universally set on the key of one sharp; the latter, generally on the natural key, or that of two sharps. In the minor mode, the former class have usually the signature of two flats, or the natural key; the latter, that of one flat. Hence the three former keys will comprise the greater part of the music in the major mode, and the three latter, of that in the minor mode, in every promiscuous collection. But if we were even to suppose each of the chords in the same key, and each of the signatures, of equally frequent occurrence, some chords would occur much oftener, as forming an essential part of the harmony of _more keys_ than others. The Vth DA, for example, forms one of the essential chords of six different keys; while the Vth G♯D♯ forms a part only of the single key of four sharps.

PROPOSITION IV.

To find a set of numbers, expressing the ratio of the probable
number of times that each of the different consonances in the
scale will occur, in any set of musical compositions.

This can be done only by investigating their actual frequency of occurrence in a collection of pieces for the instrument to be tuned, sufficiently extensive and diversified to serve as a specimen of music for the same instrument in general. This may appear, at first view, an endless task; and it would be really such, were we to take music promiscuously, and count all the consonances which the base makes with the higher parts, and the higher parts with each other. But it appears, from Prop. I. Cor. that all the positions and inversions of a chord, when the octaves are kept perfect, are equally harmonious with the chord itself. The Vth, for example, which makes one of the consonances in a common harmonic triad, is equally harmonious in its kind, with the V + VIII, which takes its place in the 3d position of this triad, and with the 4th in its second inversion. Hence, instead of counting single consonances, we have only to count chords; and this is done with the greatest ease, by means of the figures of the thorough base. The labour will be still farther abridged by reducing the derivative chords, such as the 6, the 6/4, &c. to their proper roots, as they are taken down. But even after these reductions, the labour of numbering the different chords in a sufficiently extensive set of compositions, to establish, with any degree of certainty, the relative frequency of the different signatures, would be very irksome. A method, however, presents itself, which renders it sufficient to examine the chords in such a set of pieces only as will give their chance of occurrence in _two_ keys--a major, and its relative minor.

It will be evident to all who are much conversant with musical compositions, that the _internal structure_ of all pieces in the same mode, whatever be their signature, is much the same. There is scarcely more difference, for example, in the relative frequency of different chords in the natural key, and in that of two sharps, or two flats, than there is in different pieces on the same key. If the Vth CG on the tonic has to the Vth EB on the mediant in the natural key, any given ratio of frequency _m_ : _n_, the relative frequency of the Vth DA on the tonic, and the Vth F♯ C♯ on the mediant in the key of two sharps, will not sensibly differ from that of _m_ : _n_. Hence, if we examine a sufficient number of pieces to establish the relative frequency of the different consonances in one major and its relative minor key, and, by a much more extensive investigation, ascertain the relative frequency of occurrence of the different signatures, it is evident, that by multiplying this last series of numbers into the first, and adding those products which belong to chords terminated by the same letters, we shall have a series of numbers expressing the chance of occurrence in favour of each of the consonances of the scale, when _all_ the keys are taken into view.

It was judged that 200 scores, taken promiscuously from all the varieties of music for the organ,[7] would afford a set of numbers expressing, with sufficient accuracy, the chance that a given consonance will occur in a single major, and its relative minor key. Accordingly 200 scores were examined, 150 in the major, and 50 in the minor mode, (as it will appear hereafter that this is nearly the ratio of their frequency) of the various species of music for the organ, comprising a proper share both of the simpler and of the more rapid and chromatic movements. As the selecting and reducing to their proper keys all the occasional modulations which occur in the same piece would render the labour of ascertaining the relative frequency of different signatures very tedious, it was thought best to consider all those modulations which are too transient to be indicated by a new signature, as belonging to the same key. This will account for the occurrence of the chords in the following table, which are affected by flats and sharps.

The minim, or the crotchet, was taken for unity, according to the rapidity of the movement. Bases of greater or less length had their proper values assigned them; although mere notes of passage, which bore no proper harmony, were generally disregarded. The scores were taken promiscuously from all the different keys; and were reduced, when taken down, to the same tonic; the propriety of which will evidently appear from the foregoing remarks. The following table contains the result of the investigation.

TABLE I.

+-------+---------------++---------------+---------------++-------------+
| | Common Chords.|| Flat Fifths. | 7ths. || 9-sevenths. |
| +-------+-------++-------+-------+-------+-------++------+------+
|Bases. | Major | Minor || Major.| Minor.| Major.| Minor.||Major.|Minor.|
| | mode. | mode. || | | | || | |
+-------+-------+-------++-------+-------+-------+-------++------+------+
|B III | 5 | 8 || -- | -- | 7 | -- || -- | -- |
|B | 3 | -- || 163 | 55 | 11 | 17 || 2 | -- |
|B♭ | 4 | 4 || -- | -- | -- | -- || -- | -- |
|A VII | -- | -- || -- | -- | -- | -- || 3 | -- |
|A III | 19 | 8 || -- | -- | 7 | 2 || -- | -- |
|A | 166 | 588 || 2 | 1 | 26 | 5 || 2 | -- |
|G♯ | -- | -- || 3 | 38 | -- | -- || -- | -- |
|G 3 | 18 | 15 || -- | -- | -- | -- || -- | -- |
|G | 965 | 93 || -- | -- | 178 | 15 || 3 | -- |
|F♯ | -- | -- || 46 | 4 | 11 | 2 || -- | -- |
|F | 352 | 60 || -- | -- | 11 | 12 || 7 | 3 |
|E III | 26 | 271 || -- | -- | 1 | 25 || -- | -- |
|E | 32 | 25 || 5 | 1 | 8 | -- || 1 | 4 |
|D♯ III | -- | -- || 2 | 1 | -- | -- || -- | -- |
|D♯ | -- | -- || -- | 4 | -- | -- || -- | -- |
|D III | 29 | 4 || -- | -- | 49 | 7 || -- | -- |
|D | 120 | 129 || -- | -- | 55 | 18 || 6 | 1 |
|C♯ | -- | -- || 2 | 4 | 1 | -- || -- | -- |
|C 3 | 2 | -- || -- | -- | -- | -- || -- | -- |
|C | 1769 | 275 || -- | -- | 5 | 1 || 4 | 1 |
+-------+-------+-------++-------+-------+-------+-------++------+------+

The following anomalous chords were found in the major mode, and are subjoined, to make the list complete:

8 ♯5ths on C, and 1 on D.
5 5/4ths on D, 2 on E, and 1 on G.

The left hand column of the foregoing table contains the fundamental bases of the several chords. When any number is annexed to the letter denoting the fundamental, it denotes the quality of some other note belonging to the chord. E III, for example, denotes that the various chords on E, which stand against it, have their third sharped; G 3, that the third, which is naturally major, is to be taken minor, &c. Of the two columns in each of the four remaining pairs, the left contains the number of chords belonging to each root, of the kind specified at the top, which were found in 150 scores in the major mode; and the right, the corresponding results of the examination of 50 scores in the minor mode. The diminished triad, which is used in harmonical progression like the other triads, has its lowest note considered as its fundamental. The diminished 7th, in the few instances in which it occurred, was considered as the first inversion of the 9/7th, agreeably to the French classification, and was accordingly reduced to that head.

From this table, the number of times that each consonance of two notes would actually occur, were the 200 scores played, is easily computed. We will suppose three notes, besides octaves, to be played to each chord. The octaves played it is unnecessary to take into the computation, as it would only multiply the number of consonances whose temperament is the same, in the same ratio, and would have no effect on the _ratio_ of the numbers expressing the frequency of the different consonances. In the chord of the 7th, which naturally consists of four notes, we will suppose, for the sake of uniformity, that one is omitted; and as the 7th ought always to be struck, we will suppose the Vth and IIId of the base to be omitted, each half the number of times in which this chord occurs. Considered as composed of three distinct notes, neither of which is an octave of either of the others, each chord will contain three distinct consonances. The common chord on C, for example, will contain the Vth CG, the IIId CE, and the 3d EG. The 9/7 on C will contain the VII CB, the IX, or (which must have the same temperament) the IId CD, and the 3d BD. Reducing all these consonances to their proper places, and adding those of the same name which have the same degree for their base, we obtain the following results:

TABLE II.

+--------++------------------++------------------++------------------+
| || Vths, 4ths, and || IIIds, 6ths, and || 3ds, VIths, and |
| Bases. || Octaves. || Octaves. || Octaves. |
| |+------------------++------------------++------------------+
| || Major. | Minor. || Major. | Minor. || Major. | Minor. |
+--------++--------+---------++---------+--------++--------+---------+
|B || 8 | 8 || 10 | 8 || 1141 | 214 |
|B♭ || 3 | 6 || 22 | 19 || ---- | ---- |
|A || 195 | 607 || 22 | 10 || 626 | 663 |
|G♯ || ---- | ---- || ---- | ---- || 32 | 310 |
|G || 1088 | 116 || 1090 | 125 || 22 | 23 |
|F♯ || ---- | ---- || ---- | ---- || 78 | 10 |
|F || 395 | 78 || 486 | 301 || ---- | ---- |
|E || 59 | 308 || 40 | 284 || 1828 | 308 |
|E♭ || ---- | ---- || 2 | ---- || ---- | ---- |
|D♯ || ---- | ---- || ---- | ---- || 7 | 9 |
|D || 197 | 156 || 60 | 7 || 403 | 213 |
|C♯ || ---- | ---- || ---- | ---- || 26 | 12 |
|C || 1807 | 278 || 1959 | 870 || 4 | 1 |
+========++========+=========++=========+========++========+=========+
| || 5ths, IVths, and || 7ths, IIds, and || VIIths, 2ds, and |
| Bases. || Octaves. || Octaves. || Octaves. |
| |+--------+---------++---------+--------++--------+---------+
| || Major. | Minor. || Major. | Minor. || Major. | Minor. |
+--------++--------+---------++---------+--------++--------+---------+
|B || 256 | 265 || 25 | 17 || ---- | ---- |
|B♭ || ---- | ---- || ---- | ---- || ---- | ---- |
|A || 2 | 1 || 34 | 7 || 3 | ---- |
|G♯ || 10 | 53 || ---- | ---- || ---- | ---- |
|G || ---- | ---- || 188 | 20 || ---- | ---- |
|F♯ || 74 | 7 || 1 | 2 || ---- | ---- |
|F || ---- | ---- || ---- | ---- || 17 | 16 |
|E || 10 | 1 || 20 | 27 || ---- | ---- |
|E♭ || ---- | ---- || ---- | ---- || ---- | ---- |
|D♯ || 7 | 5 || ---- | ---- || ---- | ---- |
|D || ---- | ---- || 123 | 27 || ---- | ---- |
|C♯ || 9 | 10 || 1 | ---- || ---- | ---- |
|C || ---- | ---- || 5 | 1 || 10 | 1 |
+--------++--------+---------++---------+--------++--------+---------+

Besides the following chromatic intervals:

{ 8 extreme sharp 5ths on C
Major mode. { 1 --------------------- D
{ 1 extreme flat 7th ---- G♯

{ 4 extreme sharp 6ths on F
Minor mode. { 4 extreme flat 7ths on C♯
{ 3 --------------------- G♯

It was thought best to exhibit a complete table of all the consonances which occurred in the 200 scores examined; although (Prop. II.) only the concords in the upper half of the table can be regarded in forming a system of temperament. For the more frequent consonances, this table may be regarded as founded on a sufficiently extensive induction to be tolerably accurate. For the more unfrequent chords, and especially for those which arise from unusual modulations, it expresses the chance of occurrence with very little accuracy; and it is doubtless the fact that a more extensive investigation would include some chords not found at all in this list. But it must be recollected, on the other hand, that the influence of these unusual chords on the resulting system of temperament would be insensible, could their chance of occurrence be determined with the greatest accuracy.

But none of the numbers in the foregoing table by any means expresses the chance that a given interval will occur, considering _all_ the keys in which it is found. For example, the Vth CG on the tonic of the natural key, in music written on this key, is the one of most frequent occurrence, its chance being expressed by 1807; but in the key of two flats, it becomes the Vth on the supertonic, and its chance of occurrence is only as 197. Hence the problem can be completed only by finding a set of numbers which shall express, with some degree of accuracy, the relative frequency of different signatures.

An examination of 1600 scores, comprising four entire collections of music for the organ and voice, by the best European composers, besides many miscellaneous pieces, afforded the results in the following table:

TABLE III.

+-------------+-------------+-------------+
| Signatures. | Major Mode. | Minor Mode. |
+-------------+-------------+-------------+
|4♯_s_ | 42 | 2 |
|3♯_s_ | 95 | 6 |
|2♯_s_ | 200 | 13 |
|1♯ | 322 | 72 |
|♮ | 176 | 121 |
|1♭ | 180 | 97 |
|2♭_s_ | 70 | 77 |
|3♭_s_ | 116 | 8 |
|4♭_s_ | 0 | 3 |
+-------------+-------------+-------------+
|Ratio of their sums 1201 : 399 |
+-----------------------------------------+

The chance of occurrence for any chord varies as the frequency of the key to which it belongs, and as the number belonging to the place which it holds, as referred to the tonic, in Table II., jointly. Hence the chance of its occurrence in all the keys in which it is found, is as the sum of the products of the numbers in Table III., each into such a number of Table II. as corresponds to its place in that key. To give a specimen of the manner in which this calculation is to be conducted, the numbers belonging to the major mode in the three first divisions of Table II. are first to be multiplied throughout by 176, which expresses the relative frequency of the major mode of the natural key. They are then to be multiplied throughout by 322, which expresses the frequency of the key of one sharp. But the first product, which expresses the frequency of the Vth on the tonic, now becomes GD, and must be added, not to the first, but to the fifth, in the last row of products. The product into 59, expressing the frequency of the Vth on the mediant, becomes BF♯, an interval not found among the essential chords of the natural key. In general, the products of the numbers in Table III. into those in Table II. are to be considered as belonging, not to the letters against which these multipliers stand, but to those which have the same position _with regard to their successive tonics_, as these have with regard to C. Whenever an interval occurs, affected with a new flat or sharp, it is to be considered as the commencement of a new succession of products. The IIId C♯E♯, for example, does not occur at all till we come to the key of two sharps, and even then only in occasional modulations, corresponding to the IIId on B in the natural key, whose multiplier is 10. In the key of 3 sharps it becomes another accidental chord, answering to the IIId on E in the key of C, and consequently has 40 for its multiplier. It is only in the key of 6 sharps, that it becomes a constituent chord of the key; when if that key were ever used, it would correspond to the IIId GB on the dominant of the natural key.

After all the products have been taken and reduced to their proper places, in the manner exemplified above, a similar operation must be repeated with the numbers in the second column of Table III. and those in the second columns in the three first divisions of Table II.

The necessity of keeping the major, and its relative minor key, distinct, will be evident, when we consider that the several keys in the minor mode do not follow the same law of frequency as in the major; as is manifest from the observations in Schol. Prop. III. and as clearly appears from an inspection of Table III.

But in order to discover the relative frequency of the different chords on _every_ account, the results of the two foregoing operations must be united. Now, as the numbers in the two columns of Table II. at a medium, are as 3 : 1, and those in Table III. are in the same ratio, although the factors are to each other in only the simple ratio of the relative frequency of the two modes, yet their products will, at a medium, be in the _duplicate_ ratio of that frequency. Hence, to render the two sets of results homologous, so that those which correspond to the same interval may be properly added, to express the general chance of occurrence for that interval in all the major and minor keys in which it is found, this duplicate ratio must be reduced to a simple one, either by dividing the first, or by multiplying the last series of results, by 3. We will do the latter, as it will give the ratios in the largest, and, of course, the most accurate terms. Then adding those results in each which belong to the same interval, and cutting off the three right hand figures, (expressing in the nearest small fractions those results which are under 1000) which will leave a set of ratios abundantly accurate for every purpose; the numbers constituting the final solution of the problem will stand as follows:

TABLE IV.

+--------+----------+-----------+---------+
| | Vths and | IIIds and | 3ds and |
| Bases. | 4ths. | 6ths. | VIths. |
+--------+----------+-----------+---------+
|F♯ | 67 | 29 | 1072 |
|F | 639 | 924 | 66 |
|E♯ | ---- | ---- | 12 |
|E | 548 | 323 | 1151 |
|E♭ | 265 | 363 | ½ |
|D♯ | ⅓ | ½ | 144 |
|D | 1166 | 943 | 569 |
|D♭ | 1 | 6 | ---- |
|C♯ | 25 | 12 | 581 |
|C | 816 | 1131 | 180 |
|B♯ | ---- | ---- | 4 |
|B | 221 | 135 | 1161 |
|B♭ | 418 | 654 | 5 |
|A♯ | ---- | ---- | 29 |
|A | 870 | 568 | 1085 |
|A♭ | 52 | 78 | ⅕ |
|G♯ | 5 | 4 | 365 |
|G | 1207 | 1197 | 567 |
|F♯♯ | ---- | ---- | ¼ |
|G♭ | ---- | ½ | ---- |
+--------+----------+-----------+---------+

NOTE. In this table, as well as the last, the Vths, IIIds, and 3ds are to be taken _above_, and the 4ths, 6ths, and VIths, their complements to the octave, _below_ the corresponding degrees in the first column. And, in general, whenever the Vths, IIIds, and 3ds are hereafter treated as different classes of concords, each will be understood to include its complement to the octave and its compounds with octaves.

_Scholium._

The foregoing table exhibits, with sufficient accuracy, the ratio of the whole number of times which the different chords would occur, were the 1600 scores, whose signatures were examined, actually played in succession, on the keys to which they are set, and with an instrument having distinct sounds for all the flats and sharps. Had the examination been more extensive, the results might be relied on with greater assurance as accurate; but the general similarity, not only in the structure of different musical compositions, but in the comparative frequency of the different keys in different authors; is so great, that a more extensive examination was thought to be of little practical importance.

(_To be continued._)

ART. II. _Review of an elementary Treatise on Mineralogy and Geology, being an introduction to the study of these sciences, and designed for the use of pupils; for persons attending lectures on these subjects; and as a companion for travellers in the United States of America--Illustrated by six plates. By_ PARKER CLEAVELAND, _Professor of Mathematics and Natural Philosophy, and Lecturer on Chemistry and Mineralogy in Bowdoin College, Member of the American Academy, and Corresponding Member of the Linnæan Society of New England_.

----itum est in viscera terræ:
Quasque recondiderat, Stygiisque admoverat umbris,
Effodiuntur opes----
OVID.

_Boston, published by Cummings and Hilliard, No. 1, Cornhill. Printed by Hilliard & Metcalf, at the University Press, Cambridge, New England. 1816._

This work has been for some time before the public, and it has been more or less the subject of remark in our various journals. It is, however, so appropriate to the leading objects of _this_ Journal, that we cannot consider ourselves as performing labours of supererogation while we consider the necessity, plan, and execution of the treatise of Professor Cleaveland.

An extensive cultivation of the physical sciences is peculiar to an advanced state of society, and evinces, in the country where they flourish, a highly improved state of the arts, and a great degree of intelligence in the community. To this state of things we are now fast approximating. The ardent curiosity regarding these subjects, already enkindled in the public mind, the very respectable attainments in science which we have already made, and our rapidly augmenting means of information in books, instruments, collections, and teachers, afford ground for the happiest anticipations.

Those sciences which require no means for their investigation beyond books, teachers, and study--those which demand no physical demonstrations, no instruments of research, no material specimens: we mean those sciences which relate only to the intellectual and moral character of man, were early fostered, and, in a good degree, matured in this country. Hence, in theology, in ethics, in jurisprudence, and in civil policy, our advances were much earlier, and more worthy of respect, than in the sciences relating to material things. In some of these, it is true, we have made very considerable advances, especially in natural philosophy and the mathematics, and their applications to the arts; and this has been true, in some good degree, for very nearly a century. Natural history has been the most tardy in its growth, and no branch of it was, till within a few years, involved in such darkness as mineralogy. Notwithstanding the laudable efforts of a few gentlemen to excite some taste for these subjects, so little had been effected in forming collections, in kindling curiosity, and diffusing information, that only fifteen years since, it was a matter of extreme difficulty to obtain, _among ourselves_, even _the names_ of the most common stones and minerals; and one might inquire earnestly, and long, before he could find any one to identify even _quartz_, _feldspar_, or _hornblende_, among the simple minerals; or _granite_, _porphyry_, or _trap_, among the rocks. _We speak from experience_, and well remember with what impatient, but almost despairing curiosity, we eyed the bleak, naked ridges, which impended over the valleys and plains that were the scenes of our youthful excursions. In vain did we doubt whether the glittering spangles of mica, and the still more alluring brilliancy of pyrites, gave assurance of the existence of the precious metals in those substances; or whether the cutting of glass by the garnet, and by quartz, proved that these minerals were the diamond; but if they were not precious metals, and if they were not diamonds, we in vain inquired of our companions, and even of our teachers, what they were.

We do not forget that Dr. Adam Seybert, in Philadelphia; Dr. Samuel L. Mitchill, in New-York; and Dr. Benjamin Waterhouse, in Harvard University, began at an earlier period to enlighten the public on this subject; they began to form collections; Harvard received a select cabinet from France and England; and Mr. Smith, of Philadelphia, (although, returning from Europe fraught with scientific acquisitions, he perished tragically near his native shores,) left his collection to enrich the Museum of the American Philosophical Society.

Still, however, although individuals were enlightened, no serious impression was produced on the public mind; a few lights were indeed held out, but they were lights twinkling in an almost impervious gloom.

The return of the late Benjamin D. Perkins, and of the late Dr. A. Bruce, from Europe, in 1802 and 3, with their collections, then the most complete and beautiful that this country had ever seen; the return of Colonel Gibbs, in 1805, with his extensive and magnificent cabinet; his consequent excursions and researches into our mineralogy; the commencement, about this time, of courses of lectures on mineralogy, in several of our colleges, and of collections by them and by many individuals; the return of Mr. Maclure, in 1807; his Herculean labour in surveying the United States geologically, by personal examination; and the institution of the American Journal of Mineralogy, by Dr. Bruce, in 1810;--these are among the most prominent events, which, in the course of a few years, have totally changed the face of this science in the United States.

During the last ten years, it has been cultivated with great ardour, and with great success: many interesting discoveries in American mineralogy have been made; and this science, with its sister science, Geology, is fast arresting the public attention. In such a state of things, books relating to mineralogy would of course be eagerly sought for.

No work, anterior to Kirwan, could be consulted by the student with much advantage, on account of the wonderful progress, which, within forty or fifty years, has been made in mineralogy. Even Kirwan, who performed a most important service to the science, was become, in some considerable degree, imperfect and obsolete; the German treatises, the fruitful fountains from which the science had flowed over Europe, were not translated; neither were those of the French; and this was the more to be regretted, because they had mellowed down the harshness and enriched the sterility of the German method of description, besides adding many interesting discoveries of their own. It is true we possessed the truly valuable treatise of Professor Jameson, the most complete in our language. But the expense of the work made it unattainable by most of our students, and the undeviating strictness with which the highly respectable author has adhered to the German mode of description, gave it an aspect somewhat repulsive to the minds of novices, who consulted no other book. We are, however, well aware of the value of this work, especially in the improved edition. It must, without doubt, be in the hands of every one who would be master of the science; but it is much better adapted to the purposes of proficients than of beginners.

The mineralogical articles dispersed through Aikin's Dictionary are exceedingly valuable; but, from the high price of the work, they are inaccessible to most persons.

The most recent of the French systems, that by Brongniart, seemed to combine nearly all the requisites that could be desired in an elementary treatise; and a translation of it would probably, ere this, have been given to the American public, had we not been led to expect the work of Professor Cleaveland, which, it was anticipated, would at least possess one important advantage over the work of Brongniart, and every other; it would exhibit, more or less extensively, _American localities_, and give the leading features of our natural mineral associations.

Thus it appears[8] that the work of Professor Cleaveland was eminently needed; the science, at large, needed it; and to American mineralogists it was nearly indispensable. It appeared too at a very opportune moment. Had it come a few years sooner, it might not have found many readers. Now it is sustained by the prevailing curiosity, and diffused state of information regarding mineralogy; and, in turn, no cause could operate more effectually to cherish this curiosity, and to diffuse this information still more widely, than this book. Professor Cleaveland is therefore entitled to our thanks for undertaking this task; and, in this age of book-making, it is no small negative praise if an author be acquitted of _unnecessarily_ adding to the already onerous mass of books.

With respect to the PLAN of this work, Professor Cleaveland has, with good judgment, availed himself of the excellencies of both the German and French schools.

Mr. Werner, of Fribourg, in some sense not only the founder of the modern German school of mineralogy, but almost of the science itself, is entitled to our lasting gratitude for his system of external characters, first published in 1774. In this admirable treatise he has combined precision and copiousness, so that exact ideas are attached to every part of the descriptive language, and every character is meant to be defined.

It is intended that a full description of a mineral upon this plan shall entirely exhaust the subject, and that although many properties may be found in common among different minerals, still every picture shall contain _peculiar_ features, not to be found in any other. It would certainly appear, at first view, that this method must be perfect, and leave nothing farther to be desired. It has, however, been found in practice, that the full descriptions of the Wernerian writers are heavy and dry; they are redundant also, from the frequent repetition of similar properties; and from not giving due prominence to those which are peculiar, and therefore distinctive, they frequently fail to leave a distinct impression of any thing on the mind, and thus, in the midst of what is called by the writers of this school a full _oryctognostic picture_, a student is sometimes absolutely bewildered.

Some of the modern French writers, availing themselves of Mr. Werner's very able delineation of the external characters of minerals, have selected such as are most important, most striking, distinctive, and interesting; and drawing a spirited and bold sketch, have left the minuter parts untouched: such a picture, although less perfect, often presents a stronger likeness, and more effectually arrests the attention.

This is the method of description which has been, as we think, _happily_ adopted, to a great extent by Mr. Cleaveland.

Mr. Werner, availing himself of the similarities in the external appearance of minerals, has (excepting the metals) _arranged_ them also upon this plan, without regard to their constitution; that is, _to their real nature_, or, at least, making this wholly subservient to the other: this has caused him, in some instances, to bring together things which are totally unlike in their nature, and, in other instances, to separate those which were entirely similar. Whatever may be said in favour of such a course, considered as a provisional one, while chemical analysis was in its infancy, the mind can never rest satisfied with any arrangement which contradicts the real nature of things; in a word, the composition of minerals is the only correct foundation for their classification. This classification has been adopted by several of the ablest modern French writers.

"It is believed," (says Professor Cleaveland, Preface, p. 7.) "that the more valuable parts of the two systems may be incorporated, or, in other words, that the peculiar descriptive language of the one may, in a certain degree, be united to the accurate and scientific arrangement of the other.

"This union of descriptive language and scientific arrangement has been effected with good success, by BRONGNIART, in his System of Mineralogy--an elementary work, which seems better adapted both to interest and instruct, than any which has hitherto appeared. The author of this volume has, therefore, adopted the _general_ plan of Brongniart, the more important parts of whose work are, of course, incorporated with this."

A happier model could not, in our opinion, be chosen; and we conceive that Professor Cleaveland is perfectly consistent, and perfectly perspicuous, when, adopting the chemical composition of minerals as the only proper foundation of arrangement, and, of course, rejecting the principle of Mr. Werner, which arranges them upon their external properties, he still adopts his _descriptive_ language as far as it answers his purpose. For to elect a principle of arrangement, and to classify all the members of a system so as to give each its appropriate place, is obviously quite a different thing from describing each member, after its place in a system is ascertained. In doing the latter, characters may be drawn from any source which affords them.

In his "Introduction to the Study of Mineralogy," the author has given a view at once copious, condensed, and perspicuous, of all that is necessary to be learned previously to the study of particular minerals. He begins with definitions and general principles, which are laid down with clearness.

By way of engaging the attention to the study of this department of nature, he remarks:

"From a superficial view of minerals in their natural depositories, at or near the surface of the earth, it would hardly be expected that they could constitute the object of a distinct branch of science. Nothing appears farther removed from the influence of established principles and regular arrangement, than the mineral kingdom when observed in a cursory manner. But a closer inspection and more comprehensive view of the subject will convince us, that this portion of the works of nature is by no means destitute of the impress of the Deity. Indications of the same wisdom, power, and benevolence, which appear in the animal and vegetable kingdoms, are also clearly discernible in the mineral."

"It may also be remarked," continues the author, "that several arts and manufactures depend on mineralogy for their existence; and that improvements and discoveries in the latter cannot fail of extending their beneficial effects to the aforementioned employments. In fine, the study of mineralogy, whether it be viewed as tending to increase individual wealth, to improve and multiply arts and manufactures, and thus promote the public good; or as affording a pleasant subject for scientific research, recommends itself to the attention of the citizen and scholar."

This introductory view of the importance and interest of the science cannot be charged with the fault of exaggeration, since it is most evident that neither civilization, refinement in arts, nor comfort, can exist where the properties of mineral substances are but imperfectly understood.

As regards this country, the argument admits of much amplification. The more our mineral treasures are explored, the more abundantly do they repay the research; and we trust that the period is not far distant, when we shall no longer ignorantly tread under our feet minerals of great curiosity and value, and import from other countries, at a great expense, what we, in many instances, possess abundantly at home.[9]

But to return to the plan of the author's work. Few persons, unacquainted with the science of mineralogy, would suspect that mere brute matter could exhibit many strong marks, capable of discrimination.

It may, however, be confidently affirmed, that there is no mineral which, if carefully studied, may not be distinguished by characters sufficiently decisive from every other mineral; an account of these characters ought, therefore, to precede every system of mineralogy. Professor Cleaveland has, with entire propriety, included them under the heads of crystallography, physical and external characters, and chemical characters.

He has given a clear view of the Abbé Haüy's curious discoveries regarding the six primitive figures or solids which form the bases of all crystals--the three integrant particles or molecules which constitute the primitive forms, and of the theory by which it is shown how the immensely numerous and diversified secondary or actual forms arise out of these few elementary figures.

This is certainly one of the most singular and acute discoveries of our age. It is true, there is a difference of opinion among mineralogists as to the practical use of crystallography in the discrimination of minerals. Some dwell upon it with excessive minuteness, and others seem restless and impatient of its details. The truth seems to be, that those who understand it, derive from it (wherever it is applicable) the most satisfactory aid; and it requires only a moderate knowledge of geometry to understand its principal outlines. On the other hand, it is no doubt possible, in most instances, to dispense with its aid, and to discriminate minerals by their other properties.

Of the external and physical characters of Mr. Werner, Mr. Cleaveland has given a clear account, combining into the same view the fine discriminations of the French authors, particularly regarding refraction, phosphorescence, specific gravity, electricity, chatoyement, and magnetism. The same may be said of the chemical characters. We do not know a more satisfactory and able view of the characters of minerals than Professor Cleaveland has exhibited.

We would however ask, whether, in enumerating the kinds of lustre, the term _adamantine_ should not be explained, as it is not understood by people in general, while the terms denoting the other kinds are _generally_ intelligible; whether in the enumeration of imitative forms, _lenticular_ and _acicular_ should not rather be referred to the laws of crystallization; whether _reniform_ and _mamillary_ are synonymous; whether _sandstone_, as being a mere aggregate of _fragments_, is a good instance of the _granular_ fracture; whether in its natural state (at least the common ore of nickel) is _ever_ magnetic, till _purified_, and whether cobalt is _ever_ magnetic unless _impure_.

Professor Cleaveland's remarks on _fracture_ are uncommonly discriminating and instructive, and would lead a learner to a just comprehension of this important point in the characters of minerals.

The section relating to the _chemical characters_ is concise, and professedly proceeds upon the principle of selection. It might perhaps have been, to some extent, advantageously enlarged; although, it is true, the author refers us to the particular minerals for individual instances; still it might have been well to have illustrated the general principles by a few well-chosen instances, _e. g._ how, by the blowpipe, _galena_ is distinguished from _sulphuret of antimony_; _carbonat of lead_ from _sulphat of barytes_, or _carbonat of lime_; _garnet_ from _titanium_; _plaster of Paris_ from _soapstone_, &c.; and, among trials in the moist way, how by nitric acid and ammonia, _iron pyrites_ is distinguished from _copper pyrites_; and how, by acids, _sulphat of lime_ is known from _carbonat of lime_. As the acids are used principally for trials on the effervescence of carbonats, most of which form with sulphuric acid, insoluble compounds, we should doubt whether sulphuric acid is so advantageously employed as the nitric or muriatic, in such cases, on account of the clogging of the effervescence by the thick magena, produced by a recently precipitated and insoluble sulphat.

According to our experience, the nitric or muriatic acid, diluted with two or three parts of water, is most eligible.

With respect to the blowpipe: it is _a convenience_ to have a mouth-piece of wood, or ivory, joined to a tube of metal, as Mr. Cleaveland recommends; and some authors direct to have the tube attached to a hollow ball, for the sake of condensing the moisture of the breath; but every thing which adds to the expense and complication of the instrument will tend to discourage its use; we have never found any difficulty in performing every important experiment with the common goldsmith's brass blowpipe; and are confident, that, after the learner has acquired the art, or _knack_, of propelling a continued stream of air from his mouth, by means of the muscles of the lips and cheeks, while his respiration proceeds without embarrassment through the nostrils, he will need no other instrument than the common blowpipe. Indeed it is a truly admirable instrument, instantly giving us the effect of very powerful furnaces, the heat being entirely under command, the subject of operation and all the changes in full view, and the expense and bulk of the instrument being such that every one may possess it, and carry it about his person.

The chapter on the principles of arrangement is worthy of all praise. This difficult subject is here discussed with such clearness, comprehensiveness, and candour, as prove the author to be completely master of his subject; and we are persuaded, that, on this topic, no author can be studied with more advantage. We forbear to extract, because the whole should be attentively perused in connexion, and scarcely admits of abridgement. We entirely agree with Professor Cleaveland, as we have already said, that the chemical composition of minerals is the only just foundation of their arrangement; that next in importance is the crystalline structure, including a knowledge of the primitive form, and integrant molecule; and last and least important, _in fixing the arrangement_, are the external characters: these last should be only provisionally employed, where the two first are not ascertained, or the second is not applicable. When the arrangement is once made, we _may_, however, and we commonly _shall_, in _describing_ minerals, pursue precisely the reverse order; the external characters will usually be mentioned first, the crystalline characters next, and the chemical last of all. In description, the external characters are often the most valuable; if judiciously selected and arranged, they will always prove of the most essential service, and can rarely be entirely dispensed with.

With regard to the NOMENCLATURE of minerals, we feelingly unite with Professor Cleaveland in deploring the oppressive redundancy of synonymes. Few minerals have only one name, and usually they have several. With Count Bournon we agree, that the discoverer of a mineral has the exclusive right of naming it, and that the name once given should not be changed without the most cogent reasons. What then shall we say of the ABBÉ HAÜY, of whom, whether we speak of his genius, his learning, his acuteness, his discoveries, his candour, and love of truth, or his universally amiable and venerable character, we can never think without sentiments of the highest respect and admiration? More than any modern writer he has added to the list of synonymes, often exchanging a very good name, derived perhaps from the locality or discoverer of a mineral, for one professedly significant, but connected with its subject by a chain of thought so slight, that considerable knowledge of Greek etymology, and still more explanation, is necessary to comprehend the connexion; and thus, after all, it amounts, with respect to most readers, only to the exchange of one arbitrary name for another. What advantage, for instance, has _grammatite_, alluding to a line often obscure, and still oftener wholly invisible, over the good old name _tremolite_, which always reminds us of an interesting locality; how is _pyroxene_ better than _augite_, _amphibole_ than _hornblende_, _amphigene_ than _leucite_, or _disthene_ than _sappar_. Some of the Abbé Haüy's names are, however, very happily chosen, especially where new discriminations were to be established, or errors corrected, or even a redundant crop of synonymes to be superseded by a better name. _Epidote_ is an instance of the latter, and the new divisions of the old _zeolite family_ into four species, _mesotype_, _stilbite_, _analcime_, and _chabasie_, afford a happy instance of the former. It were much to be wished, that by the common consent of mineralogists, one nomenclature should be universally adopted: for its uniformity is of much more importance than its nature.

In expressing our approbation of the principles of arrangement adopted by Professor Cleaveland, we have of course espoused those of his TABULAR VIEW, which is perhaps as nearly as the state of science will admit, erected upon a chemical basis, like that of Brongniart, to which it bears a close resemblance. Some of the subordinate parts, we could have wished had been arranged in a manner somewhat different. In the genus lime, it appears to us better to describe the species carbonat first; because, being very abundant, and its characters clear, it forms a convenient point of departure and standard of comparison, in describing the other species which have lime for their basis, and some of which are comparatively rare. The same remark we would make upon quartz, and its concomitant, pure silicious stones. There appears to us a high advantage in making these minerals clearly known first, before we proceed to those which are much more rare, and especially which are much harder, and possess the characters of gems. For example, if a learner has become acquainted with quartz, chalcedony, flint, opal, chrysoprase, and jasper, he will much more easily comprehend the superior hardness, &c. and different composition of topaz, sapphire, spinelleruby, chrysoberyl, and zircon, which we should much prefer to see occupying a later, than the first place in a tabular arrangement; and, although topaz, by containing fluoric acid, appears to be in some measure assimilated to saline minerals, it is in its characters so very diverse from the earthy salts, that we have fair reason to conclude that the fluoric acid does not stamp the character; and, as it bears so close a resemblance to the ruby and sapphire, which evidently derive their principal characters from the argillaceous earth, we perhaps ought to infer that this (the topaz,) does so too. Indeed Professor Cleaveland has sufficiently implied his own opinion, by giving these minerals a juxtaposition in his table, although the same reasons which induced the placing of the topaz next to the earthy salts, could not have justified the placing of the sapphire there. On these points we are not, however, strenuous; they are of more importance if the work be used as a text-book for lectures, than as a private companion. With respect to the _completeness_ of Professor Cleaveland's tabular view, we have carefully compared it with the third edition of Jameson's mineralogy; and although a few new species, or sub-species, and varieties have been added in this last edition, they are in general of so little importance, that Professor Cleaveland's work cannot be considered as materially deficient; and the few cases in which it is so, are much more than made up by his entirely new and instructive views of American mineralogy, to which no parallel is to be found in any other book, and which give it peculiar interest to the American, and even to the European, reader.

In another edition, (which we cannot doubt will speedily be called for,) he will of course add whatever is omitted in this, and we should be gratified to see a good article on the subject of the ærolites or stones which have fallen from the atmosphere. This subject is one, in our view, of high interest; and although _in strictness_ it may not claim a place in a tabular view of minerals, (we must confess, however, that we see no important obstacle to its being treated of under the head of native iron,) there can be no objection to its being placed in an appendix. The fall of stones from the atmosphere is the most curious and mysterious fact in natural history.