At the heart of Calculus is something called "the derivative". Never mind what that means. What is important here is that Newton wrote the symbol for the derivative as one or two little dots over the "x", as shown below. Whereas Leibniz wrote it like a fraction with a couple of "d"s. To wit:

Newton

Leibniz

Babbage turned the two notations into a metaphorical pun: "dot-age" for Newton's dots, that is, old, no longer relevant (dotage = old age senility), and "D-ism" for Leibniz's use of the letter "d", which is godly (deism).

From http://ds-wordpress.haverford.edu/bitbybit/bit-by-bit-contents/chapter-two/2-3-babbages-early-life/

Charles Babbage entered Cambridge in 1810. Hardly a bookworm, he was a charming, gregarious, and athletic young man, with a fondness for whist and sailing. Even his serious pursuits bore a lighthearted touch. During his years at Cambridge, for example, the school was caught up in a controversy over the format of the Bible. Should the book be printed with or without explanatory notes? One side sought to make the word of God more comprehensible to the masses, the other to preserve its literal purity. Cambridge, which took its religion seriously, was littered with posters and broadsides advocating one or the other side of the issue.And the rest is history! Today, although we give Newton most of the kudos for inventing Calculus, we use the Leibnizian system exclusively. Few modern mathematicians would even recognize Newton's "dots" today! And we have Charles Babbage and friends to thank for that!

At the same time, however, the university was less than zealous in its cultivation of the intellect, and the school, Newton’s alma mater and once the guiding light of European mathematics, had lost its luster. English mathematicians were trained in an inferior notation of calculus – the confusing dots of the Newtonian version as opposed to the clearly defined d’s of the Leibnizian system – and the rift between Britain and the Continent had widened to a point where most English mathematicians couldn’t decipher the publications of their Continental counterparts. English mathematics was falling by the wayside, and Babbage, Herschel, and most of the country’s bright young mathematicians and scientists were unhappy with the quality of their education.

Nothing might seem more petty and inconsequential to us today than the controversy between the dots and the d’s, but it was a significant matter in the history of science, residue of the great quarrel between Newton and Leibniz over the invention of calculus.

One spring day in 1812, Babbage picked up a broadside [leaflet] that demanded, in absurdly exaggerated terms, the publication of the unelaborated word of God. He couldn’t resist a parody. So he wrote out a plan for the establishment of a society for the propagation of “the principles of pure D-ism in opposition to the Dotage of the university.”

The satire struck a sympathetic chord with his mathematically minded schoolmates. Over the objections of the university authorities, who frowned on independent student organizations, Babbage and his friends established the Analytical Society. The group was dedicated to the overthrow of the Newtonian way, and Babbage, the intellectual rabble-rouser who founded it, was on his way to making his mark in the world.

Like most undergraduate clubs, the Analytical Society was more talk than action. It had about a dozen active members and issued only one publication, Memoirs of the Analytical Society (1813), consisting of mathematical papers written in the Leibnizian style by Babbage and Herschel, before disbanding in 1814. (Herschel, the society’s president and Cambridge’s best undergraduate mathematician, graduated in 1813 and Babbage came down the following year.) But the spirit of the group lived on. In 1816, two years after Babbage had left college, he, Herschel, and George Peacock, another ex-Analytical, launched a more mature sally against the Newtonian dots with the publication of their translation of a popular French textbook on calculus [Lacroix's "Elementary Treatise on the Differential and Integral Calculus"]. Four years later, the three men wrote a two-volume calculus workbook complete with solutions.The books accomplished what the Analytical Society had not. They were adopted by Cambridge teachers, and helped steer British mathematicians back to the mainstream.