Problems whose first solutions had been using Calculus but later was shown to be done by non-Calculus methods I was wondering about mathematical problems whose first published solutions  was obtained by using methods of Calculus but later was shown  (or known) to be solvable by using non-Calculus methods. 
Are there really any such kind of problems? 

Note:-
By the words "shown (or known)" I wanted to mean that the later non-Calculus solution were either published (and didn't existed before the solution using Calculus was published) or was later revealed to have existed even before the published solution.    
 A: I have heard that Isaac Newton, after developing calculus, did not want to use the methods in publications, so proved all about the theory of gravity using only Euclidean geometry. However, I don't have a source for this.
(I was hoping someone else would point this example out with a better source, but since no-one has, I'll post this.)
A: A beautiful example is the question, when an antiderivative is an elementary function. It was proved by Liouville in XIXth century. Methods were purely analytic.  (J. Liouville. Mémoire sur l’intégration d’une classe de fonctions transcendantes, J. Reine Angew. Math. Bd. 13, p. 93-118. (1835))
About hundred years later  computer symbolic integration begun. The methods are purely algebraic. 
For deeper studies I would recommend: Manuel Bronstein, Symbolic Integration I.
A: Semi-examples of what you ask for might be relevant for your purposes, and are easier to come up with. For example, Visual Calculus obtains results so eerily easily as to rival non-Calculus proofs. Another semi-example is when a non-Calculus solution is known, but the problem is given anyway as an exercise in Calculus. Perhaps the canonical example along this line is Regiomontanus’ angle-maximization problem. Here is the link to the Wikipedia article in which the Calculus-based solution is mentioned as being popular, but gratuitous, but with the happy result that this nifty problem has become more widely known: https://en.wikipedia.org/wiki/Regiomontanus%27_angle_maximization_problem
Also, the article, by Murray S. Klamkin, ‘On Two Classes of Extremum Problems without Calculus’ might be of interest. Here is the link to it: https://www.researchgate.net/publication/266192671_On_Two_Classes_of_Extremum_Problems_without_Calculus
