St Louis May 26th 1823.
I have taken the liberty of forwarding to you by mail one copy of a work entitled “Elements of Arithmetic”, which I have recently published. Permit me to hope that you will honor me by accepting it, and that if at some convenient time you should give it a perusal, you will have the goodness to let me know whether it meets your approbation.
I had occasion some years ago to several treatises on Arithmatic, but was generally disappointed in finding that, though most of them were perfect in a practical point of view yet, they were all or nearly all defective in their demonstrations. The transition from such treatises to any elementary works on Algebra must, in my humble opinion, be very difficult to a student who is not accustomed to make use of his own reasonings; but who, on the contrary, is in the habit of depending on what his teacher might have told him, without bringing any conviction to his mind.
This reason induced me to try to adapt to the american System, the vigour of the french and german Methods of demonstrations: I therefore prepared an elementary course of Mathematics; but doubtful whether such a plan would succeed, and living in a remote country, where all kind of difficulties must necessarily be encountered in the publication of a work of this description, I have contented myself with publishing for the present, the first part of the course, which I have even curtailed for reasons of economy; But should this plan be adopted and should you think proper to give it your sanction, I would propose to publish the whole work, consisting of:
1st The Elements of Arithmetic.
of which the copy herewith sent, though an abridgement may yet give an idea.
2d The Elements of Algebra.
Intending this, to be a continuation of the first part, it is necessary to link them, as it were, together and for that reason it begins after a few preliminary notions, by the solution of a question similar to those solved in arithmatic; but the reasonings becoming very complicated, new symbols are used to facilitate memory and equations are introduced.
These equations contain quantities represented by the alphabetical letters, those quantities must be added, subtracted &ca &ca hence the fundamental operations of arithmetic applied to liberal quantities. All these operations being once explained we proceed to the solution of equations including the quadratics, giving general ideas on the solution of those of higher degrees. The resolution of quadratic equations leads naturally to evolution, and to the calculations of quantities affected of the radical signs.
These matters, to which are added the mode of finding the greater Common measure between several quantities either numeral or literal, and the theory of continued fractions would constitute the first section.
The second would contain complete theories of Permutations, ratios, proportions, progressions and logarithms. arithmetical complements being very useful to shorten logarithmic calculations and principally to get rid of negative logarithms will be explained, this last article or chapter would be preceded by some general ideas on the formation of the tables and the manner of using them, for that purpose tables of logarithms with six decimals, of numbers from 1 to 10000, would be annexed to this volume.
The third section would contain applications of the foregoing principles to commercial transactions, such as general rules to calculate single and compound interests, discount and advance, discount or abate, annuities, perpetuities, &a all of them derived from general formula obtained in the solution of questions relative to those subjects.
The fourth section, would contain a series of questions intended for practice, and classed accordingly.
3d The Elements of Geometry.
This volume divided into chapters, is to contain all the elementary propositions demonstrated by Playfair, Butter, Simpson, Bezout, Legende &ca &ca in their excellent treatises of Geometry.
Of all elementary authors. Bezout being certainly the most intelligible for young students, I have partly followed his method and, as often as the order in which the propositions are presented, permitted it, I have preserved the demonstrations of some of these great masters.
The order of the chapters will be this:
1. Preliminary notions.
3. Angles & their measures
5. perpendicular & oblique lines, Equality of right angled triangles.
6. Theory of parallels
7. Straight lines considered relatively to the circumference of the circle, and circumferences consd relatively to each others.
8. Angles considered in the circle.
10—ditto considered relatively to the circle.
11. proportional lines
12. Similarity of triangles
13. proportional lines considered in the circle
14. Similar figures in General.
15. properties of lines drawn within a triangle.
17. mensuration of Superficies
19. comparison of Surfaces
20 squares and rectangles of lines within polygons
22 of planes.
24 mensuration of the surfaces of solids
25 relation of those surfaces.
27 solidity of bodies
28 mensuration of Solids
29 Similar & Symmetrical polyhedrons
30 comparison of Solids.
Each of the chapters to begin with the definitions of only such expressions as are used therein, hence will be so distributed as that all they suppose shall have been previously demonstrated. avoiding thus the great inconvenience generally arising by placing all of them at the beginning of the work.
Such is, Sir, the plan that I have followed and on which your opinion would be greatly gratifying.
I am confident that I intrude considerably on your time; but the desire I have to render myself useful to my country in promoting the education of the rising generation, will be my excuse before you.
I have the honor to be,very respectfully Sir,