The errors or corrections are, for purposes of interpolation, most conveniently represented graphically by a smooth curve through points with abscissas proportional to the direct readings

Aw', A, A + u', A + u' + u", &c.,

and ordinates to the corresponding corrections.

Should it be necessary to increase the accuracy by a second calibration with a thread of different length, it is only necessary to take one of approximately an integral part of (BA), and when the final curve of error is drawn, make the error at B equal to zero, distributing the difference at that point proportionally to the scale readings among the errors at the intermediate points: in other words, to shift the axis of the second curve of error so that it shall make the error at B zero. The superposition of a second curve of error deduced from the same series of observations as the first, but using another starting-point, A' differing from A by a suitable fraction of the length of the thread used, will somewhat increase the accuracy of the result by rendering interpolation more certain, but neither of these procedures is requisite except where a very detailed study of the instrument is to be made.

This method requires for each calibration the use of but a single thread. The computation is simple, and involves a minimum of approximation. Errors of observation are largely eliminated by the number of settings made in all parts of the tube, and by the inspection of the curve of lengths, both of which operations tend in an unusual degree to detect any mistakes or minor irregularities of the capillary. It avoids the common requirement of setting the thread exactly at certain definite points in the tube, or any approximate correction for slight errors in such setting, two sources of considerable error and inconvenience, especially when the thread must be set near or under a line of the graduation. And, lastly, the total time of calibration for a result of given accuracy is reduced to one half or one third of that required by Neumann's method, the quickest and most satisfactory with which I am acquainted, except that given by Pickering. The latter, described with some slight inaccuracies at the reference noted below, is a neat application of the graphical method, and the curve of lengths of thread adopted in the method which I have described is identical with the corresponding one given by Professor Pickering, while the whole process is fully one third shorter and somewhat more accurate. From a series of calibrations executed upon the same thermometer (one with a millimetre scale by Baudin of Paris), VOL. XVII. (N. S. IX.)

11

using a variety of methods, I have obtained slightly more concordant results with the proposed method than with Neumann's or Pickering's, all three possessing, however, nearly the same degree of precision, and decidedly better results with these than with any of the other existing simple methods.

Considerable aid in eliminating errors of parallax in such work is sometimes found by looking down upon the horizontal thermometer through a vertical tube having a small hole at each end. One of the cheap French microscopes with its lenses removed, and inverted in its stand, answers this purpose well. With such a device two calibrations of the above described thermometer with threads of 3 cm. and 5 cm. respectively, each with only one series of observations, and requiring not more than one hour and a half each for completion, gave results whose average difference from each other at nine points was 0.04 mm., and the arithmetical sum of the extreme differences was 0.12 mm., a result of sufficient accuracy for any class of work of which such an instrument is capable.

For brief descriptions of methods of separating threads of mercury for calibration, reference may be made to the paper by Russell and the text-book by Pickering noted below. These processes are in general use, and are safe and convenient.

KOHLRAUSCH. Physical measurements, p. 59 [Engl. Transl.].

PICKERING.

Physical Manipulation, ii. 75 (1876).

(Neumann's Method.) Carl's Rep., xv. 285 (1879).

THIESEN.

RUSSELL.

(

66

MAREK.

Carl's Repertorium, xv. 300 (1879).
[Solution by least squares.]

VON OETTINGEN. Inaug. Diss. (Dorpat, 1865).

[This I have been unable to obtain. S.W.H.]

XII.

CONTRIBUTIONS TO NORTH AMERICAN
BOTANY.

BY ASA Gray.

Presented February 8, 1882.

I. Studies of Aster and Solidago in the Older Herbaria.

ASTER and SOLIDAGO in North America, like Hieracium in Europe, are among the larger and are doubtless the most intractable genera of the great order to which they belong. In these two genera, aloug with much uncertainty in the limitation of the species as they occur in nature, there is an added difficulty growing out of the fact that many of the earlier ones were founded upon cultivated plants, some of which had already been long in the gardens, where they have undergone such changes that it has not been easy, and in several cases not yet possible, to identify them with wild originals. Late flowering Composite, and Asters especially, are apt to alter their appearance under cultivation in European gardens. For some the season of growth is not long enough to assure normal and complete development, and upon many the difference in climate and exposure seems to tell in unusual measure upon the ramification, inflorescence, and involucral bracts, which afford principal and comparatively stable characters to the species as we find them in their native haunts. I am not very confident of the success of my prolonged endeavors to put these genera into proper order and to fix the nomenclature of the older species; and in certain groups absolute or practical definition of the species by written characters or descriptions is beyond my powers. But no one has ever seen so many of the type-specimens of the species as I have, nor given more time to the systematic study of these genera. The following notes should therefore be of use.

It is noticeable that the herbarium of Nees von Esenbeck for Aster is not referred to. I cannot ascertain what has become of it. But

the types of several of his species, or specimens named by him, have been met with in other herbaria, especially in that of Lindley and that of Schultz, Bip., the latter now a part of the large collection of Dr. Cosson. As to Asters, I do not here attempt anything beyond a report of the main results of the study of certain principal herbaria ; and I leave the high northern and far western species out of the present view.

Besides general acknowledgments to the curators and botanists who have in all cases most obligingly facilitated my researches, special thanks are due to Professor Lawson, of Oxford, and Professor Eichler, of Berlin, who kindly sent to Kew, for leisurely examination and comparison, one the Asters and Solidagoes of the herbaria of Morison and of Sherard, the other those of Willdenow.

1. Notes on the North American Asters in the Older Herbaria.

I. Species of Linnæus.

A. SIBIRICUS. Founded on Gmelin's Siberian plant. Two specimens in the Linnæan herbarium: they belong to a robust form of the species which is represented in North America by the A. montanus of Richardson.

A. DIVARICATUS. Founded, as to the plant in the herbarium, on the upper part of a specimen of A. corymbosus, Ait., wanting the cordate petioled leaves, and with open inflorescence unusually foliolose. But the synonyms, both of Gronovius and of Plukenet, relate to A. infirmus, Michx., A. cornifolius, Muhl. The Linnæan name in this case should subside.

A. DUMOSUS. Herbarium specimen of the very early cultivated plant, and still in cultivation as a low and far more densely bushy plant than we find in the wild state. The figure in Hermann's Paradisus referred to by Linnæus answers well to the wild species; that of Plukenet more resembles the early cultivated form.

A. TENUIFOLIUS. This is founded upon an indigenous specimen in the herbarium, which is well described. As I have several years ago recorded, it is Nuttall's A. flexuosus, which must give place to the Linnæan name. The cited figure of Plukenet (which does not well correspond with Plukenet's phrase) belongs probably to A. polyphyllus, Willd.

A. LINARIIFOLIUS. Seemingly an indigenous specimen of this well-known species.

A. RIGIDUS.

Not in the herbarium; founded wholly on Gronovius, Fl. Virg.; and Clayton's plant is identical with the preceding species.

A. LINIFOLIUS and A. HYSSOPIFOLIUS, Mant. 114, both belonging to Galatella, an Old World group, were erroneously referred to North America (where nothing of the kind has been detected), and are to all appearance mere varieties of A. acris, L. A. linifolius originated in Hort. Cliff. No. 15, and there is a specimen in Cliffort's herbarium. The synonym of Morison relates to something else, perhaps to A. tenuifolius, L.; the plant of Gronov. Fl. Virg. referred to is A. tenuifolius, L. So that the name linifolius completely subsides, at least as regards the American flora.

A. CONCOLOR. Two specimens, one from Kalm ("K"), and perhaps the other also; probably collected in New Jersey.

A. UNDULATUS. Specimen from Kalm; the form with some cinereous pubescence, extending even to the involucral bracts; lower part of the stem wanting; pretty clearly the A. diversifolius of Michaux, and not the A. patens. The character and good figure cited from Hermann's Paradisus are a part of the foundation of the species; from his phrase, "foliis undulatis," Linnæus took the specific name; and the figure is characteristic.

A. NOVE-ANGLIE. The species is wholly clear, and comes down, with its name, from Tournefort and Hermann. But in his herbarium Linnæus had somehow confounded it with A. grandiflorus, and Smith corrected the mistake.

A. ERICOIDES. In the second edition of the Species Plantarum this is brought next to A. dumosus. The specimen in the herbarium from the Upsal Garden is an attenuated floriferous state of the received species. But the Dillenian plant from which Linnæus drew the specific name, and also the plant of Clayton, the character of which, by Gronovius, Linnæus copied as that of his A. ericoides, are A. multiflorus, Ait. Solander, therefore, ought to have continued the name of ericoides for the Dillenian and Gronovian plant, unless he could ascertain that the specimen in the Upsal Garden was in the herbarium as early as the year 1753. That cannot be done. But the two species must now continue as named and characterized in Ait. Hort. Kew.

A. CORDIFOLIUS. The species largely rests on the plants of Cornuti and of Morison, both well figured, and the latter identified in his herbarium. There is a specimen in the Linnæan herbarium, unnamed