How Did Euler Write "log"? Reading 18th Century Math

It should come as little surprise that math notation has changed in the last few centuries. Since the time of Newton and Leibniz, the operations including addition, subtraction,multiplication, division, integration, differentiation, limits, logs, sums, powers, trig functions, and so on have all changed. This adds a layer of translation to works from e.g. Euler and Lambert (1750s-1780s) which I'm working on now. Below I'll share some notation and typesetting norms from this period. 

Elementary Arithmetic

The typesetting of these symbols included variations depending on the publisher and the writer. Examples:

Arithmetic Symbols
Zero	o (lowercase O)
One	ı (small dotless i) ɪ (Latin small capital i) I (uppercase i)
Equals	a＝b (full width equals sign)
Addition	a ☩ b (cross of Jerusalem) a ✠ b (Maltese cross) a 🞣 b (Greek cross)
Subtraction	a – b (en-dash) a — b (em-dash)
Multiplication	1.2 (period) ab (adjacent symbols) a ⨉ b (times operator)
Division	\(\frac{a}{b}\)  a:b (ratio)
Exponential	\(a^b\)
Root	√a (no overbar) ∛a
Less than/greater than	a ⊰ b ('precedes under relation' symbol) a < b (sideways capital V)

Some examples from Lambert's Observationes:

x² — ax ☩ b ＝ o

( a²e ☩ b²f ☩ c²g ) : a ＝ ae  :  AbCDA.

An example from Euler's De Serie Lambertina:

x ＝ (ɪ – av)^ɪ⁄a

Sums and Logs

Two symbols are worth mentioning on their own, which are the log and the sum symbols. Euler wrote logs with a cursive l, as in:

lx ＝ v 🞣 \(\frac{1}{2}\)av² 🞣 \(\frac{1}{3}\)aa v³ 🞣 \(\frac{1}{4}\)a³v⁴ 🞣 \(\frac{1}{5}\)a⁴v⁵ 🞣 etc.

Here, lx means "log x", without a specified base. 

Euler's notation was not standard; Lambert, for example, uses 'log. x'. 

Lambert has a different notation for denoting sums as well, a notation which probably started with Leibniz. The integral symbol (long s), to Leibniz, denoted a sum of (what we would now call) infinitesimal segments. However, for sums of sequences, no 'dx' was needed, and the same symbol could be used. Thus the symbol ∫ came to be used for sums as well as integrals. 

In this notation, a sum of a sequence a, b, c, ..., g can be denoted

a + b + c + ... + g = ∫ m

Where ∫ m now represents the sum of the sequence, m being an arbitrary symbol. To represent the sum of the squares, one would write:

a² + b² + c² + ... + g² = ∫ m²

For a sum of cubes we would have ∫ m³, and so on. 

Greek Letters and Other Text

Most of the greek letters should be readily recognizable. Some, however, are a bit tricky. In particular, beta (β) is often written more like a cursive C, similar to ϐ. 

Another small bit of text that is often seen, and might be unfamiliar to some, is &c, which simply means "etc." 

References and Further Reading

My primary references for the above are Euler's De Serie Lambertina (1773, Enestrom number E532) and Lambert's Observationes variae in mathesin puram (1758). As well, Florian Cajori's A History of Mathematical Notations (2 vols., 1928-29) is essential for decoding vague or ambiguous symbols. This blog entry could hardly be a footnote in that book, which contains hundreds of citations from well known mathematicians from history, and serves as the be-all end-all reference on the history of mathematical notations. 

[The equations in this entry were formatted using Unicode symbols and MathJax.]