Origami Heaven

A paperfolding paradise

The website of writer and paperfolding designer David Mitchell

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De Viribus Quantitatis by Luca Pacioli c1496 to1508
 
The quotations on this page are from the dissertation 'Luca Pacioli and his 1500 book De Viribus Quantitatis' which can be found at http://repositorio.ul.pt/bitstream/10451/18435/1/ulfc113829_tm_Tiago_Hirth.pdf. There are links to images of the pages of the original book below each item. My thanks to Michel Grand for providing these links.

The dissertation states that De Viribus Quyantitatis was written around 1502, +/- 6 years, so between 1496 and 1508.

There are several items in the ms which bear on paperfolding history:

1. How to make a gnomon (right angle set square) without a compass.

Pacioli stresses the importance for the practical geometer to be equipped with a gnomon so “named by the philosophers” (a set-square and not the sun dial component). These are made of various materials.

For the case that such should not be available Pacioli gives instructions on how to quickly create one with a piece of paper or how to measure the right angle with a length of string, in the likes of (II.33)

To obtain a right angle with a piece of paper, fold it once to form a straight line. Next fold the line upon itself so that the two line segments are overlaid and a second straight line segment is folded, orthogonal to the first.

http://www.uriland.it/matematica/DeViribus/Pagine/404.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/405.JPG

2. The two cherries puzzle

'Take and place 2 cherries in a letter split in half. Two cherries are strung to a piece of paper cut in a particular way and are left as a puzzle to be removed. This is an impossible object. Pacioli makes reference to a missing illustration while explaining how to remove the cherries.

It is most likely that this puzzle is the following: Take a piece of paper, cut it so that an oblong rectangular slip is created. By one of its shorter sides, make a hole next to the slip of paper. The cherries, or a string with two rings attached, are placed on the slip by folding the long rectangular bit through the hole. After unfolding the piece of paper again, the stem of the cherries secures them to the strip of paper.'

http://www.uriland.it/matematica/DeViribus/Pagine/448.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/449.JPG

3. To seal a letter without any wax.

'Pacioli proposes to give instructions on how to fold a sheet of paper in such a way that it becomes its own envelope.

'Three variations are proposed for a single-sheet letter followed by instruction on how to fold an envelope and remark on a possibility to conceal a letter. The missing illustration and the volgare obscure the folds described in this section. The following descriptions seem to match those of Pacioli.'

a, 'Given a rectangular sheet of paper, one is to fold it widthwise so to obtain a strip of paper. Both ends of the paper are bent in such a way to obtain a similar trapeze standing out to each side of the strip. The side of the trapeze gives the next fold, which is to be folded over until both ends are close enough that after folding them into each other they form a square shape.'

It seems to me that this is a description of what we now know as the Love Knot Letterfold.

b, 'The second method is by far simpler. Start with a square paper. Fold it diagonally. Next, tuck one of the acute angled tips of the triangle formed into the fold of the other. The right angled tip is then tucked between them and possibly even secured by a single stitch where all of the tips overlap.'

c, 'Another method is to have the letter wrapped around a round piece of leather. It is closed. The way it is closed is not to clear. However, Pacioli, stresses that there are special tongues with rounded tips to crease the letter shut. Upon opening the crease marks will be obvious making it a hard task to restore the letter to shut state.'

It seems to me that the third method is a good description of the Chickenwire Letterfold.

http://www.uriland.it/matematica/DeViribus/Pagine/464.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/465.JPG

4. The Chinese Wallet

'Pacioli describes how to construct the device with two wooden slabs and 3 straps, forming a “wallet”. Place a straw in between the single strip and close and open the wallet again in the other possible way to produce the straw trapped by the other two straps. Pacioli tells how old people entertain infants with this device (as captured by contemporary painter Luini)'

'Pieper points out that this is the earliest known written reference of this kind of gadget, and further that Luini’s portrait was first ascribed to Leonardo. As Singmaster says, the multiple piece variant would only appear in the eighteenth century. Bossi discusses four strap variants, with crossed straps, and application of this device by Leonardo to build a theatre set.'

Ref to Bossi is either to:

Bossi, Vani (2008). “Magic Card Tricks in Luca Paciolo’s De Viribus Quantitatis” in A Lifetime of Puzzles, Taylor & Francis

or

Bossi, Vani et al. (2012). Mate-Magica I Giochi di Prestifio di Luca Pacioli, Aboca Edizioni

neither of which I have been able to refer to.

http://www.uriland.it/matematica/DeViribus/Pagine/486.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/487.JPG

5. To make small balls of white paper, as said in another part.

'Paper is coated in a mixture of dissolved Alum or sabsci (some form of soap might be meant, this would also work). The balls made of this paper are said to float.

Pacioli suggests a performance or scam where some participants are given the coated balls and others not. In the scam, those who don’t manage to make their balls float have to pay.'

http://www.uriland.it/matematica/DeViribus/Pagine/506.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/507.JPG

6. Cooking eggs, fish, meat in a paper pan.

'Pacioli describes how paper can be used as a frying pan. The paper is to be folded and closed off with pins or glue, so that it can be used as a pan. Fill it with oil. The food products are carefully placed into that oil. The pan is placed on top of a metal grid. Pacioli recommends careful usage to keep the paper from rupturing.'

http://www.uriland.it/matematica/DeViribus/Pagine/517.JPG and http://www.uriland.it/matematica/DeViribus/Pagine/518.JPG