Finish copyediting chapter 7

ch07.asciidoc

@@ -175,7 +175,7 @@ The same method for constructing a tree from four transactions can be generalize
 .A Merkle Tree summarizing many data elements
 image::images/MerkleTreeLarge.png["merkle_tree_large"]
 
-To prove that a specific transaction is included in a block, a node need only produce +log~2~(N)+ 32-byte hashes, constituting an _authentication path_ or _merkle path_ connecting the specific transaction to the root of the tree. This is especially important as the number of transactions increases, because the base-2 logarithm of the number of transactions increases much more slowly. This allows bitcoin nodes to efficiently produce paths of ten or twelve hashes (320-384 bytes) which can provide proof of a single transaction out of more than a thousand transactions in a megabyte sized block. In the example below, a node can prove that a transaction K is included in the block by producing a merkle path that is only four 32-byte hashes long (128 bytes total). The path consists of the four hashes H~L~, H~IJ~, H~MNOP~ and H~ABCDEFGH~. With those four hashes provided as an authentication path, any node can prove that H~K~ is included in the merkle root by computing four additional pair-wise hashes H~KL~, H~IJKL~ and H~IJKLMNOP~ that lead to the merkle root. 
+To prove that a specific transaction is included in a block, a node only needs to produce +log~2~(N)+ 32-byte hashes, constituting an _authentication path_ or _merkle path_ connecting the specific transaction to the root of the tree. This is especially important as the number of transactions increases, because the base-2 logarithm of the number of transactions increases much more slowly. This allows bitcoin nodes to efficiently produce paths of ten or twelve hashes (320-384 bytes) which can provide proof of a single transaction out of more than a thousand transactions in a megabyte sized block. In the example below, a node can prove that a transaction K is included in the block by producing a merkle path that is only four 32-byte hashes long (128 bytes total). The path consists of the four hashes H~L~, H~IJ~, H~MNOP~ and H~ABCDEFGH~. With those four hashes provided as an authentication path, any node can prove that H~K~ is included in the merkle root by computing four additional pair-wise hashes H~KL~, H~IJKL~ and H~IJKLMNOP~ that lead to the merkle root. 
 
 [[merkle_tree_path]]
 .A Merkle Path used to prove inclusion of a data element