Charles Darwin died in 1882 but his grandfather, Erasmus Darwin, actually proposed a theory of evolution in 1770. Charles was strongly influenced by his scientific grandfather.
Thomas Henry Huxley coined 'Darwinism' in 1860. A scientific definition of Darwinism is taken from 'Life' book - Nature Library - Evolution. The full title of Darwin's first book states his own definition; On The Origin Of Species by means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. His second book was, The Descent of Man, and Selection in Relation to Sex (1871).
But Darwin did not know about DNA and the role of genes, so evolutionary theory today has... evolved accordingly!
In 1856 Austrian monk Gregor Johann Mendel launched the first of a
series of experiments that were to demonstrate that inheritance, like
evolution, is not a chaos or chance, or miracle, but a matter of law.
Darwin never heard of Mendel's work, and the monk's reports lay
ignored by the scientific world for decades.
Others contributed to the development of evolutionary theory. Hugo de Vries (d. 1935) developed the first mutation theory through extensive studies of the evening primrose. He thought he had formed new species through mutations. (He had actually identified segregated characters, not mutations but his work provided a platform on which to build.) De Vries discovered the import of Mendel's work and, in a paper read before the German Botanical Society in 1900, he gave Mendel full credit for one of the most momentous discoveries in scientific history.
It did not take long for Darwin's take on evolutionary theory to be adopted, but it has been constantly adapted! There is not a whole lot of Darwin's original interpretation still in vogue today, due to on-going adaptations that take into consideration new discoveries. Darwinian theory started off a change in direction regarding scientific beliefs about the origins of species, which was quickly and enthusiastically accepted by many, but what we have today has evolved significantly from his (and his grandfather's) initial thinking. But given that Thomas Henry Huxley coined 'Darwinism' in 1860, your textbook quote has reason to claim general acceptance within a decade.
However, there were many who did not agree with the new 'Darwinism'. It was by no means universally accepted within a decade of Darwin's first book being published. It's likely, however, that the clamour of excitement generated by this new enthusiasm received much coverage in the media of the day, so that it would appear that virtually the whole scientific world was running after the idea.
EDIT - This compilation is partly from 'Life' series - Nature Library - 'Evolution' by Ruth Moore and The Editors of LIFE, 1964, published by Time-Life Int. It is also partly from a paper on the subject of Evolution that I put together a few years ago. I am a published author myself, but not on this subject, and my own document is just one in my own private collection.