Was Julius Caesar originally spelled with and I before "J" was invented? Or was it spelled some other way? If so, how?
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As others stated, on monumental inscriptions, the name of Julius Caesar would look similar to
However, saying it was "spelled with an I instead of a J" may be misleading, because 'J' as a later innovation did not arise from thin air: while 'I' and 'J' were not distinguished in Roman times, they existed as graphically distinct variants of the same letter, which always looked more like 'I' in capitals, but could sometimes look more like a (dotless) 'ȷ' in everyday cursive script.
The distinction was originally just a matter of natural variation within people's handwriting, but in time, a habit tended to form where the first and/or last letters in a word may come to stand out more, resulting in 'ȷ' being used more often than 'ı' in those contexts, just like a better-defined 'v' would stand out more than a more fluidly-written 'u'. I think this pattern can be observed in a few of the Vindolanda tablets for example. I consider it somewhat natural for the first bit of handwriting to be written more carefully or incisively than what follows.
In the case of 'ȷ' (often called the equivalent of "long I" in several modern languages), the distinction may also have been influenced by the standardized classical Roman habit of writing a longer 'I' to indicate that it was a long vowel, something they routinely did in inscription too, and which was unique to 'I' as the same indication was given for other vowel letters by writing an apex above them, at least when useful to reduce ambiguity. This use, in any case, is distinct from the specific shape and use that 'J' later evolved into.
Another letter that often underwent shape/length changes depending on position in Roman cursive was 's', and this distinction also survived into modern times as the long ſ, this time used within words whereas 's' would be used at the extremities. Since this also appears in Caesar's name in both word positions, we can reconstruct the way his name would typically have been written in Roman cursive by approximating it, at least in concept, with the modern lowercase form
Perhaps because words that begin with 'I' or 'V' in Latin are statistically more likely to use those as the semivowels /j/ and /w/, rather that the vowels /i/ and /u/ which are more common in the middle of words, eventually — but well after Roman times, and partly after the Middle Ages — 'J' and 'V' established themselves as semivocalic forms, while 'I' and 'U' remained for the vowels, and since the informal cursive distinction in "glyph length" became systematized as swashes in printing, this was no longer just restricted to handwriting.
In the meanwhile, the phoneme /w/ had since long mutated into /β/ or /v/ in Latin and its successors, so 'V' is rarely used as a semivowel in today's languages that employ the Latin script, instead representing [v] or related fricatives or approximants. In some languages, /j/ stopped being a semivowel too, becoming [ʒ] in French for instance. So, the way 'V' and 'J' are used today varies with languages, and additionally, Germanic languages that still had a separate semivocalic /w/ created a convention of using a double 'U' or 'V' (which were not yet generally distinguished at that stage) to reduce the ambiguity with both /u/ and /v/.
A tangentially related point is the origin of the dot on both 'i' and 'j' in modern non-capital scripts: this can be traced back to the apex diacritic mentioned above, which eventually became the acute accent used in many modern languages to mark either word stress or vowel quality resulting from ancestral length distinction, although some languages still use it to mark length. Because 'ı' and 'ȷ' were thin letters which often caused confusion (especially with 'm' when they were next to 'n') in minuscule handwriting, the apex or acute accent, then known as a tittle, started being systematically used regardless of length, quality or stress, merely to make those letters stand out as separate from the adjoining ones.