Connection through ethernet shield is possible but not when using MQTT on my arduino

Question

I have been trying to send messages via MQTT from my Arduino to my amazon web server. The code below connects the ethernet client but not the MQTT client. Why would my MQTT client not be connecting?

#include <SPI.h>
#include <Ethernet.h>
#include <PubSubClient.h>

byte mac[]    = { 0x12, 0x42, 0x98, 0x85, 0x49, 0x3A }; //MAC address of server
char server[] = "http://52.1.29.117/"; //web address of server
IPAddress ip(172, 31, 51, 13); //IP address of server

void callback(char* topic, byte* payload, unsigned int length) {
  // handle message arrived
}

EthernetClient ethClient;
PubSubClient client(server, 80, callback, ethClient);

void setup()
{
    Serial.begin(9600);
    while (!Serial) {
        ; // wait for serial port to connect. Needed for Leonardo only
    }

    // start the Ethernet connection:
    if (Ethernet.begin(mac) == 0) {
        Serial.println("Failed to configure Ethernet using DHCP");
        Ethernet.begin(mac, ip);
    }
    delay(1000);
    Serial.println("connecting...");

    if (ethClient.connect(server, 80)) {
        Serial.println("connected");
        // Make a HTTP request:
        ethClient.println("GET /search?q=arduino HTTP/1.1");
        ethClient.println("Host: www.google.com");
        ethClient.println("Connection: close");
        ethClient.println();
    }