The phrase to give (someone) furiously to think means: to set (someone) thinking very hard or seriously, to give (someone) much food for thought.
This phrase is a Gallicism, translating French donner furieusement à penser.
For example, the English author of detective fiction Agatha Christie (1890-1976) made the Belgian detective Hercule Poirot use this phrase in The Mysterious Affair at Styles. A Detective Story (New York: John Lane Company – London: John Lane, The Bodley Head – 1920):
“If it had been Inglethorp who was carrying on an intrigue with Mrs. Raikes, his silence was perfectly comprehensible. But, when I discovered that it was known all over the village that it was John who was attracted by the farmer’s pretty wife, his silence bore quite a different interpretation. It was nonsense to pretend that he was afraid of the scandal, as no possible scandal could attach to him. This attitude of his gave me furiously to think, and I was slowly forced to the conclusion that Alfred Inglethorp wanted to be arrested. Eh bien! from that moment, I was equally determined that he should not be arrested.”
The English lexicographer and grammarian Henry Watson Fowler (1858-1933) condemned this and other Gallicisms in A Dictionary of Modern English Usage (Oxford: The Clarendon Press – London: Humphrey Milford – 1926):
To use Gallicisms for the worst of all reasons—that they are Gallicisms—, to affect them as giving one’s writing a literary air, to enliven one’s dull stuff with their accidental oddities, above all to choose Gallicisms that presuppose the reader’s acquaintance with the French original, these are confessions of weakness or incompetence. If writers knew how ‘leap to the eye’ does leap to the eye of the reader who, in dread of meeting it, casts a precautionary glance down the column, or how furious is the thinking that ‘give furiously to think’ stirs in the average Englishman, they would leave such paltry borrowings alone for ever.
The earliest occurrences of the phrase to give (someone) furiously to think that I have found are as follows, in chronological order:
1-: From French Literature, published in The Saturday Review of Politics, Literature, Science, and Art (London, England) of Saturday 28th December 1889—the following is about Shylock (Paris: Charpentier), by E. Haraucourt:
M. Haraucourt is, we doubt not, a clever man. But when you write a play “d’après Shakspeare,” the following insertion in the Trial scene gives furiously to think:—
[…]
Let us say nothing against this in its kind. But, if any composition of any human mind could be not even a thousand miles “après” Shakspeare, this is.
2-: From the review of Poems (New York: Anson D. F. Randolph and Co., 1889), by Harriet McEwen Kimball—review published in the Catholic Champion (New York City, New York, USA) of February 1890:
The peculiarity wherein these pseudo-poets differ from other artists lies in the fact that when the latter fail in their special department they try some other or cease producing, whereas the former never weary of rushing into print with their pseudographs and nothing stops them but the flinty-hearted publisher or their own lean purse. Why there should be this persistency is a problem difficult of solution. As the French say, it “gives furiously to think.”
3-: From French Literature, published in The Saturday Review of Politics, Literature, Science, and Art (London, England) of Saturday 9th August 1890—the following is about Petits Lundis (Paris: Perrin), by Antonin Bunand:
We do not honestly think that he knows much more of “Les poètes modernes de l’Angleterre” than M. Sarrazin (whom he is reviewing) has told him, and it gives most furiously to think that he quotes Tieck and Novalis among “contemporaries and successors” of Heine.
4-: From The Colonies & India and American Visitor: A Weekly Journal of General Information, With Especial Reference to Colonial, Indian, and American Interests (London, England) of Saturday 10th January 1891:
Some really delightful examples of English as she is wrote are supplied by a translated edition of La Couturière, which is published in the United States. The following passage, apparently on the question of ladies’ riding costume, will, as the talented translator would probably say, “give furiously to think” those who want to know what Paris has to say on the matter:—“The very traveling America has admitted the plan of adopting trousers and forsake the skirt. Here some have much laughed and made fun of. However, the idea has grown and made a long way for there is at New York a group of charming and young ladies most rich and elegant who keep that up. Only misses and ladies have not fallen in the exaggeration that we believe: that will not oblige us to become androgynus. They are not willing that we leave our graces or charm so as to cut out hair and so become missed boys; we shall remain thorough women but a special costume is looked for traveling, trunting, racing—so as the amazone has been created exclusively for riding; the base of that costume will naturally be the trousers; but which one. Will it be short held in gaiters with a small skirt like certain hunting fitting? . . . Will they be inspired from the oriental puffing trousers who are more full and more discret than our present skirts. That is the question. I thought I should not leave unknown the revolution they are doing above the ocean. Let us not be ignorant of the symptoms. When the moment will come the french dress-maker will be as uswal at the heighth of the circumstances.”
5-: From Literature and the Drama, by William Archer, published in The New Review (London, England) of July 1893—the following is about Questions at Issue (London: William Heinemann, 1893), by Edmund Gosse:
More than one piece among this pleasant baker’s dozen, by its subject as well as by its handling, intends staying. Perhaps only one, “An Election at the English Academy,” challenges that mansion which awaits critical studies that have intrinsic interest as pieces of literature independent of subject; and a very amusing piece of literature it is. (But who was the elderly man of pleasant appearance who was talking to Lord Salisbury with spectacles on? He will “give furiously to think” to the scholiast of the thirtieth century, will that elderly man.)