*A Treatise on Linear Differential Equations*(1889). We give extracts from the Preface of this work, together was an extract from a review at THIS LINK.

Here we give extracts from Prefaces of two other books by Thomas Craig: *Wave and Vortex Motion* (1879), and *A Treatise on Projections* (1880). We also give a couple of notes to put some of Craig's references into context.

**1. Wave and Vortex Motion, by Thomas Craig.**

**Preface.**

The subject of Hydrodynamics embraces many of the most difficult problems in the range of physical research.

Although, at all times attracting the attention of the greatest minds, it is only within little over a century past that much real progress has been made in the solution of the many and complicated cases presented by the ordinary phenomena of Fluid Motion. The names of Euler, Lagrange and Laplace in the last century, and of Helmholtz, Stokes, Thomson, Rayleigh and Kirchhoff in this, stand out predominantly as those that have done the most to advance the theory to its present position. The object of the following article is to present in a short space the more important points in the Mathematical Theory of Fluid Motion, as it has been developed by these investigators. It is a want severely felt by any one making a study of this subject, that there exists no separate and complete treatise on Hydrodynamics.

It is a fact, I think, greatly to be regretted, that the men who do the most for the real advancement of science so seldom present to the world the result of their labours and extensive knowledge, in any other form than an occasional memoir in a scientific journal, or in a communication to a learned society. There are, however, notable exceptions to this general rule, as witness: Maxwell's treatise on Electricity and Magnetism; Rayleigh on Sound; Cayley's Elliptic Functions, and a few others. If some one would present to the public a treatise on Hydrodynamics, of the scope of those mentioned on other subjects, he would certainly receive the gratitude of all physical students, and confer a great boon upon the scientific world.

Thomas Craig

Baltimore

April 1879.

**2. A Treatise on Projections, by Thomas Craig.**

**Preface.**

In the following paper an attempt has been made to give a sufficiently comprehensive account of the theory of projections to answer the requirements of the ordinary student of that subject. The literature of projections is very large, and its history presents the names of many of the most eminent mathematicians that have lived between the time of Ptolemy and the present day. In the great mass of papers, memoirs, etc., which have been written upon projections there is much that is of the highest value and much that, though interesting, is trifling and unimportant. Thus many projections have been devised for map construction which are merely elegant geometrical trifles. Although in what follows the author has taken up every method of projection with which he is acquainted, he has not thought it necessary in the cases referred to to do more than mention them and give references to the papers or books in which they may be found fully treated.

As the different conditions which projections for particular purposes have to satisfy are so wholly unlike, it is necessary, of course, to have a different method of treatment for the various cases. Thus no general theory underlying the whole subject of projections can be given. Perhaps the only division of the subject - omitting the simple case of perspective projection - that has ever been fully treated is that of projection by similarity of infinitely small areas. This is a most important case, the general theory of which, for the representation of any surface upon any other, has been given by Gauss. The mathematical difficulties in the way of such a treatment of equivalent projections and projections by development seem to be insurmountable, but certainly offer a most attractive field for mathematical research. The author has attempted to add a littel to what is already known on these subjects, but feels that what he has done is of little consequence unless, indeed, it should tempt some abler mathematician to take up the subject and develop it as it deserves. A few of the solutions of simple problems in the paper, it is believed by the author, are new and simpler than any he was able to find in the writings of others. The solution of the problem of the projection of an ellipsoid of three unequal axes upon a sphere by Gauss's method is also believed to be new. With these few exceptions there is no claim of originality in what follows: the attempt having simply been made to present in as simple and natural form as possible what others have done. The two treatises on projections from which much aid has been obtained are those by Littrow and Germain. Littrow's 'Chorographie', which appeared in Vienna in 1833, was at that time a most valuable work, but is at the present day too limited in its scope to be of much use to the student. Unquestionably the most important treatise on the subject at this time is Germain's "Traité des projections," which contains an account of almost every projection that has ever been invented. The author is under much obligation to this work, both for references to original sources and for solutions of particular problems. In cases where processes or diagrams are taken from this work that are by the author supposed to have been original with M. Germain, special mention is made in the text; when, however, Germain has drawn from earlier sources no mention is made of his book, but as far as possible references to the original papers are given. The opening brief chapter on conic sections has been taken in great part from Salmon's 'Conic Sections'. The object of that chapter is only to give in a simple manner some of the more important and elementary properties of the curves of the second order, so that convenient reference could be made in the subsequent part of the paper to the various formulas connected with these curves, and also simple means given for constructing them. At the request of Superintendent Carlile P Patterson the paper has been divided into two parts. The first part contains the mathematical theory of projections, while the second part contains merely such a sufficient account of the various projections as will enable the draughtsman to construct them. ...

This paper is a résumé in vary compact form of most that is of importance in the subject of projections together with a comparison of the principal methods of projection in use at the present day. The paper forms Appendix No. 15 in the annual report of the Coast and Geodetic Survey for 1880. In conclusion, the author may say that although he has endeavoured to give full credit to previous writers on the subject, still it is possible that some reference has been omitted. This should, however, be taken as an unintentional oversight, or due to the fact that the author has not been able to trace back to its original source the solution of process in question, and not in any case to a desire to withhold from any other author his full measure of credit.

Thomas Craig

Coast and Geodetic Survey Office

19 August 1880.

**Note 1.**

*Traité des projections des cartes geeographiques; repreesentation plane de la sphère et du sphéroide* was written by Adrien Adolphe Charles Germain (1837-1895)and published in 1866. Adrien Germain studied at the École Polytechnique and describes himself on the title page of his book as 'Ingénieur Hydrographe de la Marine' and 'Membre de la Societé de Géographie'.

**Note 2.**

*Chorographie; oder Anleitung, alle Arten von Land* by Joseph Johann von Littrow (1781-1840) was published in 1833. Littrow was an astronomer at the University of Vienna who taught, among others, Nikolai Brashman and Joseph Raabe.