Short Answer: Because Prophet Muhammad came AFTER the Bible was written. Though he isn’t mentioned by name, there are numerous prophecies of a “Spirit of Truth” who will come after Jesus, “repeating only what he has heard from God.” Many such prophecies point to Prophet Muhammad.
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You have rightly said that the Quran mentions the names of a number of prophets mentioned in the Bible (both Jewish and Christian), but the Bible does not mention the name of Prophet Muhammad (PBUH).
The obvious reason is that Muhammad (PBUH) came AFTER the Bible was written.
That is to say, all the Biblical prophets including Noah, Moses, David, and Jesus, lived before Muhammad (PBUH).
In the long chronological line of prophets, Muhammad (PBUH) was the Last.
So Muhammad’s story could not have been included in the Bible among the stories of the prophets who had lived before the Bible came into being.
He Was Prophesied, Though
All the same, one can see clear prophecies about the advent of Muhammad (PBUH) both in the Old Testament and the New Testament of the Bible.
Here is an example from the Old Testament:
I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth; and he shall speak unto them all that I shall command him. (Deuteronomy 18:18)
This is what Jehovah tells Moses about the Prophet to come.
From this verse we may understand the following:
(1) The Prophet to come would be from among their brethren – i.e from the brothers of the Children of Israel. The brothers of the Children of Israel are the Children of Ishmael; so the Prophet will be one of the Children of Ishmael.
(2) That Prophet will be “like unto thee”, i.e like Moses. Hence there will be a lot of similarities between that Prophet and Moses.
(3) God will put His “words in his mouth”- i.e he will repeat the exact words of God as revealed to him.
We can find that all the above aspects of the Prophet to come were realized in Muhammad (PBUH).
Here is an example from the New Testament:
I have much more to say to you, more than you can now bear. But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come […] (John 16:12-13)
Here we see how Jesus was bidding farewell to his disciples.
He tells them that he won’t be with them for long; and that he hasn’t conveyed the whole truth to them as they won’t be able to bear it at that time.
But when the Spirit of truth comes, (he said) “he will guide you into all the truth.”
It is also said that the Spirit of truth speaks only what he hears; i.e what he hears from God.
The above prophecy also is fulfilled in Muhammad the Last Prophet of God as “Spirit of Truth”.
For this reason, even though Muhammad is not actually mentioned by name in the Old and the New Testament, his coming is foretold there.
Islam: The Completed Revelation Which Began with Moses
From the point of view of Islam, Jews and Christians are People of the Book, namely the people who received revealed scriptures from God, such as the Torah, the Psalms and the Gospel.
The Guidance of God came to the people through these books revealed to the respective prophets, until God completed it through the Quran revealed to the Last of the Prophets, Muhammad (PBUH).
The earlier scriptures were limited in scope, and have no history of having a single authentic canon guarded safely from interpolation or corruption.
Whereas the Quran as perfected scripture is kept safeguarded from corruption or interpolation in its original language of Arabic from the moment its first verse was revealed to the End Times.
This is because, the Quran is sent by God as Guidance for the whole of humanity for all time.
(From Ask About Islam’s archives)