I well remember how Jem used to go to Dr. Jennison’s school in a very old house under the Washington elm; and later to Mr. Whitman at the Hopkins Academy. There I once went with him. I remember old Dr. Hedge who lived near by with two maiden daughters, and the Dixwells. I remember old Mrs. Lowell, and Miss Louisa Greenough who lived in a house facing the Common, and her bringing me a bottle of Stuart’s syrop. I remember going to see Mr. Whittemore’s factory. They either made or used curry combs there. Whittemore soon failed and was made postmaster. Later he was clerk in a drygoods shop. I remember being weighed in Deacon Brown’s store in a huge pair of scales.
I remember Dr. Gould coming to my father to take lessons, and he left college in 1844.
I remember driving with father to Dr. Bache’s camp at Blue Hill, in 1845. I well remember the Quincys who left Cambridge in 1845. I also remember a great deal about the building of the New House. I recollect the discussion of the plans before it was begun, and the building of it. Then it was not finished in time, and the Bartletts who were to take the old house moved in and we boarded with them, and John Bartlett made a wonderful trick cabinet of drawers out of cigar boxes. I remember eating my first meal in the new house of oysters, seated on the back stairs. I think we must have moved in 1845. I remember in that year Judge Story’s funeral, the Irish Famine, the Mexican War, and the retirement of Professor Treadwell. I soon after began going to Miss Ware’s school.
I remember early visits to my grandmother in Salem, to different houses. One time in the cars with Aunt Lizzie, Miss Margaret Fuller was with us, and had a book with pictures about an imp in a bottle. She impressed me a good deal. She moved to N.Y. in 1844 and in 1846 left this country and was drowned on the return voyage with her husband, the Marquis Ossoli, in 1850, which event of course I remember the talk about.
13
Note on Pythagorean Triangles
| c. 1890 | Houghton Library |
A Pythagorean triangle is a set of 3 integer numbers proportional to the legs and hypotheneuse of a right triangle. It is irreducible if the 3 integers have no common measure. The number of irreducible Pythagorean triangles of which a given number is hypotheneuse is 0, if the number contains a prime factor not of the form (4n + 1); otherwise, it is equal to 2p − 1 where p is the number of its different prime factors. For example, 725 = 52 · 29. Accordingly,
It is also true that (500)2 + (525)2 = (725)2; but that triangle is reducible. Again, 1105 = 5 · 13 · 17. Accordingly,
14
Hints toward the Invention of a Scale-Table
| c. 1890 | Houghton Library |
[Version 1]
§1. A system of logarithms is a system of numbers corresponding, one-to-one, to natural numbers in such a way that pairs of natural numbers which are in the same ratio to one another have logarithms which differ from one another by the same amount. Thus, since
10:15 = 14:21
it follows that
log 10 − log 15 = log 14 − log 21.
Logarithms were invented by Napier, 1614.
§2. A logarithmic scale is a scale on which natural numbers are set down at distances from the origin measured by their logarithms. If we apply a piece of paper to such a scale and mark off the distance of 15 from 10 and measure this on from 14 we shall find 21; thus solving the proportion.
The logarithmic scale was invented by Edmund Gunter, 1624.
§3. All linear logarithmic scales are similar. Consequently, different systems of logarithms are only different scales of measurement along a logarithmic scale.
§4. Suppose I convert the edge of this sheet into a rude logarithmic scale, using the spaces between the lines as units of measurement. If I have no means at hand of subdividing them, except that of writing the numbers regularly, the proper subdivision of the scale may be treated in the use of it as a separate problem.
Practice with this scale will suggest several patentable inventions. You will see the scale is at the same time a table of antilogarithms, that is of numbers corresponding to given logarithms.
§5. How long must the slip of paper be for use with this scale?
§6. With how many places of figures can this scale be usefully inscribed? If too many are used the subdivisions of one space will not be in equal proportion throughout the space. Thus, if the last numbers were instead of 670, 741, 819, 905
6700190
7405684
8185466
9047342
the differences would be
705494
779782
861876
952658
and the second differences would be
74288
82094
90782.
The intervals evidently could not be subdivided proportionally.
But if not enough places were inscribed, the table would lose very much of its utility.
§7. You will notice that the first differences are very nearly (though not exactly) tenths of the means between successive pairs of numbers. This can evidently be put to use in subdividing the intervals.
Then what should be the number of spaces on the scale-table?
§8. Suppose you have the problem As 25 is to 55 so is 28 to the answer, how do you proceed? The roughest use of the scale gives 61; but how to get the next figure? We have
The second sheet (R 221:3) of the first version of “Hints toward the Invention of a Scale-Table.” Peirce used the vertical space between the lines (present but invisible in the image above) to define a unit of measurement for the three-place logarithmic scale inscribed along the left-hand margin of the sheet. (By permission of the Houghton Library, Harvard University.)
This is very nearly
This is precisely right.
Required to multiply 23 by .0434. This is
The true answer is 0.9982.
Had we inscribed the scale with an additional figure, we should have had