Monday 24 January 2011

Notes on the History of Graph Paper

Just re-ordered graph paper for next year for my department. We don't use nearly as much these days as five or ten years ago... calculators have made them much less common in schools. It reminded me that I hadn't actually put anything here about my notes on the history of graph paper, so for those who are interested...

Graph paper, a math class staple, was developed between 1890 and 1910. During this period the number of high school students in the U.S. quadrupled, and by 1920, according to E.L Thorndike, one of every three teenagers in America “enters High School”, compared to one in ten in 1890. The population of “high school age” people had also grown so that the total number of people entering HS was six times as great as only three decades before. Research mathematicians and educators took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.” When the University of Chicago opened in 1892 E.H. Moore was the acting head of the mathematics department. “Moore was born in Marietta, Ohio, in 1862, and graduated from Woodward High School in Cincinnati. “(from Milestones in (Ohio) Mathematics, by David E. Kullman) Moore was President of the American Mathematical Society in 1902. The Fourth Yearbook of the NCTM, Significant Changes and Trends in the Teaching of Mathematics Throughout the World Since 1910, published in 1929, has on page 159, “The graph, of great and growing importance, began to receive the attention of mathematics teachers during the first decade of the present century (20th)” . Later on page 160 they continue, “The graph appeared somewhat prior to 108, and although used to excess for a time, has held its position about as long and as successfully as any proposed reform. Owing to the prominence of the statistical graph, and the increased interest in educational statistics, graphic work is assured a permanent place in our courses in mathematics.” [emphasis added]
Hall and Stevens “A school Arithmetic”, printed in 1919, has a chapter on graphing on “squared paper”.

Using google's n-gram viewer I arrived at the conclusion that from 1880 until appx 1925 the term square paper was the most popular with coordinate paper close behind.  The earliest mention of graph paper in relation to math education was in Advanced Algebra by Joseph Victor Collins.  I found a use in 1890 of logarithmic graph paper (defined as "Ordinate scales printed on logarithmic graph paper" in a report " Geological Survey Water-supply Paper - Issues 1890-1894").  The term may have been more used in engineering fields before it was adapted in mathematician.  However the shift occurred, by 1830 "Graph paper" was the most common term, and by 1940 it was more common than the other two terms combined, and by 1960 it reached eight times the usage of either of the others.  *PB 

John Bibby has written (August,2012) to advise me that John Perry, who was at the time President of the Institution of Electrical Engineers, has a section on "Use of Squared Paper" in an article in Nature in 1900 (The teaching of mathematics, Nature Aug 1900 pp.317-320.) They wanted $19 to see the article, so I take John at his word. I did find another similar endorsement of "squared paper" by Perry in "Englands Neglect of Science" published in 1900 also. On page 18 after several lamentations about trained engineers who had no ability/understanding of the mathematics applying to their field, he writes: "I tell you, gentlemen, that there is only one remedy for this sort of thing. Just as the antiquated method of studying arithmetic has been given up, so the antiquated method of
studying other parts of mathematics must be given up. The practical engineer needs to use squared paper." 

The actual first commercially published “coordinate paper” is usually attributed to Dr. Buxton of England in 1795 (if you know more about this man, let me know). The earliest record I know of the use of coordinate paper in published research was in 1800. Luke Howard (who is remembered for creating the names of clouds.. cumulus, nimbus, and such) included a graph of barometric variations. [On a periodical variation of the barometer, apparently due to the influence of the sun and moon on the atmosphere. Philosophical Magazine, 7 :355-363. ]
[The above was gathered from a numbur of authoritive sources including a Smithsonian site, but on a recent visit to Monticello, the home of my longtime favorite American Prisident, Thomas Jefferson, I discovered it was in error. I found a use by Jefferson in his use of the paper for architectural drawings earlier than any of these dates. Here is the information from the Moticello web site.]
Prior to 1784, when Jefferson arrived in France, most if not all of his drawings were made in ink. In Paris, Jefferson began to use pencil for drawing, and adopted the use of coordinate, or graph, paper. He treasured the coordinate paper that he brought back to the United States with him and used it sparingly over the course of many years. He gave a few sheets to his good friend David Rittenhouse, the astronomer and inventor:

"I send for your acceptance some sheets of drawing-paper, which being laid off in squares representing feet or what you please, saves the necessity of using the rule and dividers in all rectangular draughts and those whose angles have their sines and cosines in the proportion of any integral numbers. Using a black lead pencil the lines are very visible, and easily effaced with Indian rubber to be used for any other draught." {Jefferson to David Rittenhouse, March 19, 1791}
A few precious sheets of the paper survive today.
The increased use of graphs and graph paper around the turn of the century is supported by a Preface to the “New Edition” of Algebra for Beginners by Hall and Knight. The book, which was reprinted yearly between the original edition and 1904 had no graphs appearing anywhere. When the “New Edition” appeared in 1906 it had an appendix on “Easy Graphs”, and the cover had been changed to include the subhead, “Including Easy Graphs”. The preface includes a strong statement that “the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shews that anything on a smaller scale (such as ‘millimeter’ paper) is practically worthless in the hands of beginners.” He finishes with the admonition that, “The growing fashion of introducing graphs into all kinds of elementary work, where they are not wanted, and where they serve no purpose – either in illustration of guiding principles or in curtailing calculation – cannot be too strongly deprecated. (H. S. Hall, 1906)” The appendix continued to be the only place where graphs appeared as late as the 1928 edition. The term “graph paper seems not to have caught on quickly. I have a Hall (the same H S Hall as before) and Stevens, A school Arithmetic, printed in 1919 that has a chapter on graphing on “squared paper”. Even later is a 1937 D. C. Heath text, Analytic Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase “coordinate paper” (page 223, topic 153). Even in 1919 Practical mathematics for Home Study by Claude Irwin Palmer introduced a section on “Area Found by the Use of Squared Paper” and then defined “paper accurately ruled into small squares” (pg 183). It may be that the term squared paper hung on much longer in England than in the US. I have a 1961 copy of Public School Arithmetic (“Thirty-sixth impression, First published in 1910) by Baker and Bourne published in London that still uses the term “squared paper” but uses graphs extensively.

Of course "graph paper" could not have preceded the term "graph" for a curve of a function relationship, and many teachers and students might be surprised to know that it was not until 1886 when George Chrystal wrote in his Algebra I, "This curve we may call the graph of the function." The actual first known use of the term "graph" for a mathematical object actually predates this event by only eight years and occurred in a discrete math topic.   J. J. Sylvester published a note in February 1878 using 'graph' to denote a set of points connected by lines to represent chemical connections. In that note "Chemistry and Algebra", Sylvester
wrote: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph" .

A more or less famous Kekule structure is the benzine shown at right.

This short note in Nature was more a notice of the more complete paper he had written in American Journal of Mathematics, Pure and Applied, would appeared the same month,  "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices,"  The term "graph" first appears in this paper on page 65.  The images he uses appear below. 

Graph has come to have multiple meanings in mathematics, but for most students it relates to the graph of functions on the coordinate axes.  The origin is from the Greek graphon, to write, perhaps with earlier references to carving or scratching. Jeff Miller's web site suggests that the use of graph as a verb may have first been introduced as late as 1898. 
In a post to a history newsgroup, Karen Dee Michalowicz commented on the history of graphing:
It is interesting to note that the coordinate geometry that Descartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later.  In fact, I find it appearing in the mid 1800's in my old college texts in analytic geometry.  It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. [This matches rise of  graph paper in the same periods].  Graphing is most often found in books by Wentworth.  Even so, the texts written in the 20th century, perhaps until the 1960's, did not all have graphing.  Taking Algebra I in the middle 1950's, I did not learn to graph until I took Algebra II

Math historian Bea Lumpkin has written about the early graphs by the Egyptians in what was an early use of what painters call the grid method:
In my article ... I suggest, "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square.  It needs just one short, important step from the use of square grids to the location of points by coordinates.  
In the same posting she comments on the finding of graphs in Egyptian finds dating back to 2700 BC: 
"An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara.  It is a limestone flake, apparently complete, measuring about 5 x 7 x 2 inches, inscribed on one face in red ink, and probably belongs to the IIIrd dynasty"  Here is the reason that Clark and Engelbach attached great importance to the diagram.  It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally.  The vertical coordinates are given in cubits, palms and fingers.  The horizontal spacing, the authors write "... most probably that is to be understood as one cubit, an implied unit elsewhere."  To clinch their analysis, Clarke and Engelbach observe:  "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated tot he curve obtained from the data on the ostrakon. 

This certainly lays claim to the oldest line graph I have ever heard.  


Nona said...

Graph paper ... now that is one of the few very fond memories I have of math classes. However, if I'd had a "cool" teacher like you, perhaps I would have enjoyed math a lot more. But seriously, "squared paper" just sounds so ... I don't know ... SQUARE!

Pat's Blog said...

Well, we math folk sometimes are a little square...

John Bibby said...

Perry (1900:319) has section on "Use of Squared Paper" - See Bibby (1986, p.80) "History of Teaching Statistics" and

Perry (1900)The teaching of mathematics Nature Aug 1900 pp.317-320