# Differences between modern and old mathematical notations

Note: I didn't write the word "ancient" in the title because I want to see the notation from 1400 A.D. to 1700 A.D.
Mathematical notation has changed very much from the past millennium, and new notations are made for new concepts. For example, there was a time when the notations for representing numbers was different in different regions. I saw somewhere a piece of a page from one of Euler's books or articles:

This is very close to modern notation, but not same as it (for example, the root symbols are larger). After seeing this page, a question cam in my mind:

What are some major differences in today's notation and notation of 1400 A.D. to 1700 A.D. era?

These differences are to be mentioned:

• Differences in symbols representing numbers, for example, somewhere in an old article (probably of Euler's) the number $$1$$ was written as $$I$$ (I thought that it was roman, but 2 was written normally)
• Major differences in "minor notations". For example, while reading Ramanujan's handwritten manuscripts, I found that He wrote $$\&c$$ instead of the normal $$\dots$$ in infinite sum. He would have written: $$\frac{1}{1^2}+\frac{1}{2^2}+\&c$$ instead of: $$\frac{1}{1^2}+\frac{1}{2^2}+\dots$$
• Differences in the "texture" of the symbols. For example, in the image I have given above, it can be noticed that the integral symbols and the root symbols are a little bit different.
• Differences in terminology. Many examples are in the first answer, and I think there are very few that he didn't mention. If there are any more, please also mention them.

Like most big list questions, please mention only one difference per answer. If I missed any other difference in symbols case, please mention in the comments or edit the question.

• See F.Cajori's A history of mathematical notations: you can find (quite) everything in it. – Mauro ALLEGRANZA Nov 10 '20 at 6:55
• Besides Cajori's book, in recent years one can also look at older literature yourself without digging through university library stacks for it, and I would encourage you to take advantage of this. For example, see the various old journals freely available you can find here (in the past 15 years I've managed to obtain nearly complete runs of these and many other journals from the 1800s). For titles of journals to search for, see the list on pp. ix-xlix here. – Dave L Renfro Nov 10 '20 at 15:18
• Let $\pi_0$ denote the value we now denote $\pi$. In Euler's proofs, $\pi$ could denote $\pi_0/2$, $\pi_0$ or $2\pi_0$, whichever was more convenient. – J.G. Nov 12 '20 at 6:57

# I. Notes on notation and English terms in 1800s literature (primarily related to series)

The following is from the introductory comments for something involving conditionally convergent series I was working on several years ago.

1. The logarithm function is often denoted by a lower case L, sometimes italicized (e.g. Catalan's book) and sometimes not italicized (e.g. Laurent's book). Incidentally, this leads to many instances of ln appearing, but these denote the logarithm of $$n$$ rather than the often now used logarithm function $${\ln}.$$ Iterated logarithms were denoted either by consecutive L's or the use of superscripts (e.g. $$\ln \ln \ln n$$ might be denoted by llln or lll n or $${l^3}n,$$ or by similar expressions in which the lower case L is NOT italicized).

2. $$n!$$ (factorial) was often denoted by placing $$n$$ within a "partial box" formed by using only the left and bottom sides of a rectangular box. Anytime you are unsure how factorials are denoted in a book, simply look in places where power series expansions of exponential and trigonometric functions appear. However, most of the time a "factorial notation" was not used, and instead the first few terms (and the general term, if not obvious) were written.

3. Multiplication was often denoted by a period (easier to typeset than our current centered dot notation), and thus the expression $$1.2.3.4$$ represents $$4! = 24.$$

4. Sometimes an overline bar (called a vinculum) is used instead of parentheses for grouping purposes.

5. Miscellaneous terms:

vulgar fraction = a quotient of integers written in vertical fraction format

impossible number = non-real complex number

possible number = real number

finite value/quantity = (especially before late 1800s) nonzero value/quantity

absolute term = constant term (e.g. of a polynomial)

numerically greater = absolute value is greater

algebraically greater = greater as signed numbers

contrary signs = opposite signs

vanishing fraction = $$0/0$$ indeterminate form

integral rational function = polynomial

integral function = power series

development = series expansion

line = curve (e.g. not necessarily a straight line)

right line or straight line = line

curve of $$n$$ dimensions = curve of algebraic degree $$n$$

rectilinear asymptote = linear/oblique asymptote

invariable quantity = constant quantity

squared paper = graph paper

the term be wanting = the term is absent

equation is rigorous = exact equality holds

rigid proof = rigorous proof

1. Colloquial journal names:

Annales de Gergonne = Annales de Mathématiques Pures et Appliquées

Crelle's Journal and Borchardt's journal = Journal für die reine und angewandte Mathematik

Darboux's Bulletin = Bulletin des Sciences Mathématiques et Astronomiques [“et Astronomiques” was later dropped]

Grunert's Archiv = Archiv der Mathematik und Physik

Hoffmann's Journal = Zeitschrift fur Mathematischen und Naturwissenschaftlichen Unterricht

Liouville's Journal = Journal de Mathématiques Pures et Appliquées

Terquem's journal = Nouvelles Annales de Mathématiques

# II. Glossary of early to mid 20th century terms in some nowhere differentiable continuous functions literature:

The following list is from near the beginning of this mathoverflow answer.

enumerable = countable

unenumerable = uncountable

progressive derivative = right derivative

regressive derivative = left derivative

non-dense = nowhere dense

everywhere dense = dense in the smallest interval containing the set (sometimes “dense”, without being prefaced by “everywhere”, means in context “everywhere dense”)

incrementary ratio = the ordinary difference quotient that a derivative is the limit of

line of invariability (of a function) = an interval on which the function is constant

the set exists = the set is not empty.