Bentley, William, 1759-1819. Papers of William Bentley, 1783-1815: an inventory. Letter from James Winthrop to William Bentley, 1788 July 19. HUG 1203.5 Box 1, Harvard University Archives.

Cambridge 19 July 1788——

Rev. Sir, By Mr. Goodwin who will pass thro’ Salem to-morrow, I have an opportunity to send you a memorandum of the principles on which the pentagraph is constructed.

Let AB, BC, DE, & EF be four equal [diagram] rulers put together by pins so that they may freely move round the points of intersection: & let the opposite rulers be parallel & their segments equal, that is let EG = HB = GC, & let GB = EH = AH. Then will the point E be always in the line AC, & as AH is to AB so is AE to AC, for in every dilatation or contraction of AC, the angle AHE will always equal ABC, & their reflective chords AE & AC will be of necessity proportional to AH & AB.

If therefore A be made a central point to the whole machine, & the index is placed at C, & a pencil at E (in the room of the pin which commonly keeps the middle rulers together) & the index be conducted over the line of an original plan, the pencil will mark out a copy without any further trouble, & the copy will be in all its parts to the original as AE to AC, & on the other hand if C be made the central point & the index be placed at A, the effect will only be to alter the proportion between the original & the copy, but without the changing or destroying the proportion of parts in the copy. If the index be placed at E, & the pencil at one of the extremes, the proportion Between one of the the copy & the original will be reversed, but the & the copy will be the largest, but will retain the Symmetry of its parts. If the index be placed at one extreme & the pencil at the other, & E be made the central points, the copy & original will be as the Segments of the line AC; but this is not necessarily done except when the copy is to be of the Same Size as the original, in which case all the rulers are bisected, AH must equal HB, EG = GF, CG = GB, & EH = HD. Then AE will equal EC, & consequently each point A & C being equally distant from E, if this last be made the central point, each of the extremes will have an equal motion about it, & therefore the copy will equal the original.

I think these remarks include all the cases that can happen. The method is theoretically exact, & I believe in practice as precise as any,

The usual method of measuring distances with the compasses is also in theory correct; but when we come to join the points found in that way, by the motion of a persons hand, it will wil be found to deviate far from the truth. This generally, you know, obliges us in the first trial to make use of a pencil, & after many corrections we came near the truth. But with the pentagraph all the labor of finding points by measuration is avoided, & altho’ from the quivering of a person's man’s hand it is impossible to keep the index all the time upon the lines of the original plan, a person must be careful indeed, who should fail of coming so near the truth in the copy, as that he should not be able to correct it without measuring.

In copying plans of buildings & other right-lined figures, I generally after laying aside the pentagraph sharpen the point of a pencil & with my ruler straiten all the lines, before I blacken them with the steel pen. After the pen has been used, if the whole paper be rubbed with white bread or elastic gum, all the marks of the pencil will be effaced, & the ink remain uninjured.

The pentagraph I find to be a most useful instrument for drawing ellipses, but tho’ the application is easy it would require time to ex- go -plain thro' the demonstration of it, & it would after all be of anyno use to you. Since I completed the construction of it, I find one in used which consists of five rulers, & probably was, from the name, the original invention. The annexed figure will give you an idea of it. The rulers are all to be equal. The four outside ones are fastened together by pins at the ends, so that the Shape of the parallelogram may be altered at pleasure. BD is the fifth ruler & is in like manner secured at the ends so as to be parallel to two sides of the parallelogram, but is removeable at pleasure [diagram] along the sides FG & AE. A being the central point the pencil is to be placed at C in the fifth ruler BC being always equal to AB. The index being at G, the copy will be to the original as AC to AG. The same variations will take place as in the pentagraph first described. Its only fault is that it consists of five parts instead of four, & therefore is more complicated than is necessary.

Yo'u see how large a letter I have written. Excuse the quantity. I would have made it Shorter if I had known how.

I wish you would contrive to make me another visit. The last was not long enough.

Yours affectionately James Winthrop

Rev. Mr. Bentley—

Winthrop July 19, 1788.